% Encoding: UTF8
@COMMENT{BibTeX export based on data in FAU CRIS: https://cris.fau.de/}
@COMMENT{For any questions please write to crissupport@fau.de}
@article{faucris.117409864,
abstract = {We investigate the dynamics of unimodal maps $f$ of the interval restrictedto the omega limit set $X$ of the critical point for cases where $X$ is aCantor set. In particular, many cases where $X$ isa measure attractor of $f$ are included. We give two classes of examples ofsuch maps, both generalizing unimodal Fibonacci maps [LM,BKNS]. In allcases $f{\_}{X}$ is a continuous factor of a generalized odometer (an addingmachinelike dynamical system), and at the same time $f{\_}{X}$ factors ontoan irrational circle rotation. In some of the examples we obtain irrationalrotations on more complicated groups as factors.},
author = {Bruin, Hendrik Pieter and Keller, Gerhard and St. Pierre, Matthias},
doi = {10.1017/S0143385797086392},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {12671287},
peerreviewed = {Yes},
title = {{Adding} machines and wild attractors},
volume = {17},
year = {1997}
}
@article{faucris.330645930,
abstract = {We derive a straingradient theory for plasticity as the Γlimit of discrete dislocation fractional energies, without the introduction of a coreradius. By using the finite horizon fractional gradient introduced by Bellido et al. (Adv Calc Var 17:1039–1055, 2024), we consider a nonlocal model of semidiscrete dislocations, in which the stored elastic energy is computed via the fractional gradient of order 1α. As α goes to 0, we show that suitably rescaled energies Γconverge to the macroscopic straingradient model of Garroni et la. (J Eur Math Soc (JEMS) 12:1231–1266, 2010).},
author = {Almi, Stefano and Caponi, Maicol and Friedrich, Manuel and Solombrino, Francesco},
doi = {10.1007/s00208024030206},
faupublication = {yes},
journal = {Mathematische Annalen},
keywords = {26A33; 35R11; 49J45; 74C05},
note = {CRISTeam Scopus Importer:20241101},
peerreviewed = {Yes},
title = {{A} fractional approach to straingradient plasticity: beyond coreradius of discrete dislocations},
year = {2024}
}
@article{faucris.121723184,
abstract = {
Inspired by an example of Grebogi et al (1984 Physica D 13 261–8), we study a class of model systems which exhibit the full twostep scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold (1998 Random Dynamical Systems (Berlin: Springer)). The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point.
The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measurepreserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.
},
author = {Anagnostopoulou, Vasiliki and Jäger, Tobias and Keller, Gerhard},
doi = {10.1088/09517715/28/7/2587},
faupublication = {yes},
journal = {Nonlinearity},
peerreviewed = {Yes},
title = {{A} model for the nonautonomous {Hopf} bifurcation},
url = {http://iopscience.iop.org/article/10.1088/09517715/28/7/2587},
volume = {28},
year = {2015}
}
@article{faucris.114224044,
abstract = {Let $W{\_}{lambda,b}(x)=sum{\_}{n=0}^inftylambda^n g(b^n x)$ where $bgeqslant2$ is an integer and $g(u)=cos(2pi u)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Bar'ansky, B'ar'any and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W{\_}{lambda,b}$ equals $2+frac{loglambda}{log b}$ for all $lambdain(lambda{\_}b,1)$ with a suitable $lambda{\_}b<1$. This reproduces results by Bar'ansky, B'ar'any and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multiscale estimate.
},
author = {Keller, Gerhard},
faupublication = {yes},
journal = {Annales de l'Institut Henri Poincaré  Probabilités Et Statistiques},
pages = {169181},
peerreviewed = {Yes},
title = {{An} elementary proof for the dimension of the graph of the classical {Weierstrass} function},
volume = {53},
year = {2015}
}
@incollection{faucris.119438924,
abstract = {
We study infinite systems of globally coupled maps with permutation invariant interaction as limits of large finitedimensional systems. Because of the symmetry of the interaction the interesting invariant measures are the exchangeable ones. For infinite systems this means in view of de Finetti's theorem that we must look for time invariant measures within the class of mixtures of spatial Li.d. processes. If we consider only those invariant measures in that class as physically relevant which are weak limits of SRBmeasures of the finitedimensional approximations, we find for systems of piecewise expanding interval maps that the limit measures are in fact mixtures of absolutely continuous measures on the interval which have densities of uniformly bounded variation.
The law of large numbers is violated (in the sense of Kaneko) if a nontrivial mixture of i.i.d. processes can occur as a weak limit of finitedimensional SRBmeasures. We prove that this does neither happen for C3expanding maps of the circle (extending slightly a result of Jiirvenpiiii) nor for mixing tent maps for which the critical orbit finally hits a fixed point (making rigorous a result of Chawanya and Morita).
},
author = {Keller, Gerhard},
booktitle = {Fractal Geometry and Stochastics II},
doi = {10.1007/9783034883801{\_}9},
editor = {Christoph Bandt, Siegfried Graf, Martina Zähle},
faupublication = {yes},
note = {UnivISImport:20150309:Pub.2000.nat.dma.pma29.anergo},
pages = {183208},
peerreviewed = {unknown},
publisher = {Springer},
series = {Progress in Probability},
title = {{An} ergodic theoretic approach to mean field coupled maps},
volume = {46},
year = {2000}
}
@article{faucris.117410524,
author = {Keller, Gerhard},
doi = {10.1016/S03044149(97)000732},
faupublication = {yes},
journal = {Stochastic Processes and their Applications},
pages = {187206},
peerreviewed = {Yes},
title = {{A} new estimator for information dimension with standard errors and confidence intervals},
volume = {71},
year = {1997}
}
@article{faucris.122855744,
abstract = {Let f be a nonrenormalizable Sunimodal map. We prove that f is a ColletEckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map.},
author = {Keller, Gerhard},
faupublication = {yes},
journal = {Colloquium Mathematicum},
pages = {229233},
peerreviewed = {unknown},
title = {{A} note on dynamical zeta functions for {S}unimodal maps},
volume = {84/85},
year = {2000}
}
@article{faucris.117545384,
abstract = {We extend Filippova's result on weak convergence of v. Mises' functionals and prove a weak invariance principle. Applications toUstatistics are given and extensions to contiguity and weakly dependent processes are briefly discussed.},
author = {Denker, Manfred and Keller, Gerhard and Grillenberger, Christian},
doi = {10.1007/BF01897813},
faupublication = {no},
journal = {Metrika},
pages = {197214},
peerreviewed = {Yes},
title = {{A} note on invariance principles for v.\ {Mises}' statistics},
volume = {32},
year = {1985}
}
@article{faucris.110519464,
abstract = {For a class of quasiperiodically forced timediscrete dynamical systems of two variables (θ, x) ∈ duoblestruct T sign^{1} × ℝ+ with nonpositive Lyapunov exponents we prove the existence of an attractor Γ̄ with the following properties: 1. Γ̄ is the closure of the graph of a function x = φ(θ). It attracts Lebesguea.e. starting point in duoblestruct T sign^{1} × ℝ+. The set {θ : φ(θ) ≠ 0} is meager but has full 1dimensional Lebesgue measure. 2. The omegalimit of Lebesguea.e. point in duoblestruct Tsign^{1} × ℝ+ is Γ̄, but for a residual set of points in duoblestruct Tsign^{1} × ℝ+ the omega limit is the circle {(θ, x) = 0} contained in Γ̄. 3. Γ̄ is the topological support of a BRS measure. The corresponding measure theoretical dynamical system is isomorphic to the forcing rotation.},
author = {Keller, Gerhard},
faupublication = {yes},
journal = {Fundamenta Mathematicae},
note = {UnivISImport:20150305:Pub.1996.nat.dma.pma29.anoteo},
pages = {139148},
peerreviewed = {Yes},
title = {{A} note on strange nonchaotic attractors},
volume = {151},
year = {1996}
}
@article{faucris.315836690,
abstract = {We devise a new technique to prove twodimensional crystallization results in the square lattice for finite particle systems. We apply this strategy to energy minimizers of configurational energies featuring twobody shortranged particle interactions and threebody angular potentials favoring bondangles of the square lattice. To each configuration, we associate its bond graph which is then suitably modified by identifying chains of successive atoms. This method, called stratification, reduces the crystallization problem to a simple minimization that corresponds to a proof via slicing of the isoperimetric inequality in ℓ^{1} . As a byproduct, we also prove a fluctuation estimate for minimizers of the configurational energy, known as the n^{3 / 4} law.},
author = {Friedrich, Manuel and Kreutz, Leonard},
doi = {10.1007/s10955023032027},
faupublication = {yes},
journal = {Journal of Statistical Physics},
keywords = {Atomic interaction potentials; Crystallization; Edge isoperimetric inequality; Square lattice; Stratification},
note = {CRISTeam Scopus Importer:20231222},
peerreviewed = {Yes},
title = {{A} {Proof} of {Finite} {Crystallization} via {Stratification}},
volume = {190},
year = {2023}
}
@article{faucris.311094445,
abstract = {We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed EulerianLagrangian formulation, as magnetizations are defined on the deformed set in the actual space. We consider lowenergy configurations by rescaling the elastic energy according to the linearized von K'arm'an regime. First, we identify a reduced model by computing the Gammalimit of the magnetoelastic energy, as the thickness of the plate goes to zero. This extends a previous result obtained by the first author in the incompressible case to the compressible one. Then, we introduce applied loads given by mechanical forces and external magnetic fields, and we prove that sequences of almost minimizers of the total energy converge to minimizers of the corresponding energy in the reduced model. Subsequently, we study quasistatic evolutions driven by timedependent applied loads and a rateindependent dissipation. We prove that energetic solutions for the bulk model converge to energetic solutions for the reduced model, and we establish a similar result for solutions of the approximate incremental minimization problem. Both these results provide a further justification of the reduced model in the spirit of the evolutionary Gammaconvergence.},
author = {Bresciani, Marco and Kru, Martin},
doi = {10.1137/21M1446836},
faupublication = {yes},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {dimension reduction; EulerianLagrangian energies; evolutionary Gammaconvergence; Gammaconvergence; magnetoelasticity; rateindependent processes},
note = {CRISTeam Scopus Importer:20230929},
pages = {31083168},
peerreviewed = {Yes},
title = {{A} reduced model for plates arising as lowenergy gammalimit in nonlinear magnetoelasticity},
volume = {55},
year = {2023}
}
@article{faucris.111532564,
abstract = {
We
},
author = {Fabbri, Roberta and Jäger, Tobias and Johnson, Russel and Keller, Gerhard},
faupublication = {yes},
journal = {Topological Methods in Nonlinear Analysis},
note = {UnivISImport:20150309:Pub.2005.nat.dma.pma29.ashark},
pages = {163188},
peerreviewed = {Yes},
title = {{A} {Sharkovskii}type theorem for minimally forced interval maps},
year = {2005}
}
@incollection{faucris.118982424,
abstract = {We study onedimensional lattices of weakly coupled piecewise expanding interval maps as dynamical systems. Since neither the local maps need to have full branches nor the coupling map needs to be a homeomorphism of the infinite dimensional state space, we cannot use symbolic dynamics or other techniques from statistical mechanics. Instead we prove that the transfer operator of the infinite dimensional system has a spectral gap on suitable Banach spaces generated by measures with marginals that have densities of bounded variation. This implies in particular exponential decay of correlations in time and space.},
address = {Heidelberg},
author = {Keller, Gerhard and Liverani, Carlangelo},
booktitle = {Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics},
doi = {10.1007/11360810{\_}6},
faupublication = {yes},
isbn = {9783540242895},
note = {UnivISImport:20150420:Pub.2005.nat.dma.pma29.aspect},
pages = {115151},
peerreviewed = {unknown},
publisher = {Springer},
series = {Lecture Notes in Physics},
title = {{A} spectral gap for a onedimensional lattice of coupled piecewise expanding interval maps},
volume = {671},
year = {2005}
}
@article{faucris.306949331,
abstract = {We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of Bfree subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz Bfree systems. Bull. Pol. Acad. Sci. Math. 65(2) (2017), 139152]. Additionally, we provide an example of a Bfree Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 of S. Ferenczi et al [Sarnak's conjecture: what's new. Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics (Lecture Notes in Mathematics, 2213). Eds. S. Ferenczi, J. KułagaPrzymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163235].},
author = {Dymek, Aurelia and Kasjan, StanisłAW and Keller, Gerhard},
doi = {10.1017/etds.2023.43},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
keywords = {automorphisms group; Bfree dynamics; sets of multiples; Toeplitz dynamical system; trivial centralizer},
note = {CRISTeam Scopus Importer:20230630},
peerreviewed = {Yes},
title = {{Automorphisms} of {B}free and other {Toeplitz} shifts},
year = {2023}
}
@article{faucris.117400844,
abstract = {We study bifurcations of invariant graphs in skew product dynamical systems driven by hyperbolic surface maps T like Anosov surface diffeomorphisms or baker maps and with onedimensional concave fibre maps under multiplicative forcing when the forcing is scaled by a parameter r=e^{t}. For a range of parameters two invariant graphs (a trivial and a nontrivial one) coexist, and we use thermodynamic formalism to characterize the parameter dependence of the Hausdorff and packing dimension of the set of points where both graphs coincide. As a corollary we characterize the parameter dependence of the dimension of the global attractor A{\_}t: Hausdorff and packing dimension have a common value dim(A{\_}t), and there is a critical parameter t{\_}c determined by the SRB measure of T^{1} such that dim(A{\_}t)=3 for t < t{\_}c and t > dim(A{\_}t) is strictly decreasing for t{\_}c < t < t{\_}{max}.},
author = {Keller, Gerhard and Otani, Atsuya},
doi = {10.1080/14689367.2013.781267},
faupublication = {yes},
journal = {Dynamical SystemsAn International Journal},
pages = {123139},
peerreviewed = {Yes},
title = {{Bifurcation} and {Hausdorff} dimension in families of chaotically driven maps with multiplicative forcing},
volume = {28},
year = {2013}
}
@article{faucris.202482898,
abstract = {We consider skew product dynamical systems f:Θ×ℝ→Θ×ℝ,f(휃,y)=(T휃,f휃(y))" role="presentation">f:Θ×ℝ→Θ×ℝ,f(𝜃,y)=(T𝜃,f𝜃(y)) with a (generalized) baker transformation T" role="presentation">T at the base and uniformly bounded increasing C3" role="presentation">C3 fibre maps f휃" role="presentation">f𝜃
with negative Schwarzian derivative. Under a partial hyperbolicity
assumption that ensures the existence of strong stable fibres for f" role="presentation">f,
we prove that the presence of these fibres restricts considerably the
possible structures of invariant measures — both topologically and
measure theoretically, and that this finally allows to provide a
“thermodynamic formula” for the Hausdorff dimension of set of those base
points over which the dynamics are synchronized, i.e. over which the
global attractor consists of just one poin},
author = {Keller, Gerhard and Otani, Atsuya},
doi = {10.1142/S0219493718500090},
faupublication = {yes},
journal = {Stochastics and Dynamics},
keywords = {Skew product, negative Schwarzian, attractor, Hausdorff dimension},
peerreviewed = {Yes},
title = {{Chaotically} driven sigmoidal maps},
volume = {18},
year = {2017}
}
@article{faucris.117444624,
abstract = {We prove a simple formula for the zetafunction of coded systems generated by circular codes (and more generally by circular Markov codes). We apply this to the loop counting method for determining the topological entropy of a subshift of finite type, to the zetafunction of the Dyckshift over 2N symbols, and to the zetafunction of a subshift of finite ype which is obtained from a full shift by deleting one block of arbitrary length.},
author = {Keller, Gerhard},
doi = {10.1016/00973165(91)90023A},
faupublication = {yes},
journal = {Journal of Combinatorial Theory Series A},
pages = {7583},
peerreviewed = {Yes},
title = {{Circular} codes, loop counting, and zetafunctions},
volume = {56},
year = {1991}
}
@article{faucris.112990724,
author = {Keller, Gerhard},
faupublication = {yes},
journal = {Colloquium Mathematicum},
note = {UnivISImport:20150309:Pub.2004.nat.dma.pma29.comple},
pages = {7376},
peerreviewed = {unknown},
title = {{Completely} mixing maps without limit measure},
volume = {100},
year = {2004}
}
@article{faucris.120798744,
abstract = {We consider families of dynamics that can be described in terms of PerronFrobenius operators with exponential mixing properties. For piecewise C ^{2} expanding interval maps we rigorously prove continuity properties of the drift J(λ) and of the diffusion coefficient D(λ) under parameter variation. Our main result is that D(λ) has a modulus of continuity of order O(δλ.(logδλ) ^{2}), i.e. D(λ) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are quantified numerically for the latter class of maps by using exact series expansions for the transport coefficients that can be evaluated numerically. We numerically observe strong local variations of all continuity properties. © 2008 IOP Publishing Ltd and London Mathematical Society.},
author = {Howard, Phil J. and Keller, Gerhard and Klages, Rainer},
doi = {10.1088/09517715/21/8/003},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2008.nat.dma.zentr.contin},
pages = {17191743},
peerreviewed = {Yes},
title = {{Continuity} properties of transport coefficients in simple maps},
volume = {21},
year = {2008}
}
@article{faucris.111272744,
abstract = {We present an approach to the investigation of the statistical properties of weakly coupled map lattices that avoids completely cluster expansion techniques. Although here it is implemented on a simple case we expect similar strategies to be applicable in a much larger class of situations.},
author = {Keller, Gerhard and Liverani, Carlangelo},
faupublication = {yes},
journal = {Discrete and Continuous Dynamical Systems},
keywords = {Coupled map lattice; Exponential decay of correlations; Spatiotemporal chaos; Spectral gap; Transfer operator},
note = {UnivISImport:20150309:Pub.2004.nat.dma.pma29.couple},
pages = {325335},
peerreviewed = {Yes},
title = {{Coupled} map lattices without cluster expansion},
volume = {11},
year = {2004}
}
@incollection{faucris.117412284,
abstract = {We describe the transfer operator approach to coupled map lattices (CML) in cases where the local map is expanding but has no Markov partition (e.g. a general tent map). The coupling is allowed to be nonlocal, but the total influence of all sites j<F NaN> 6= i on site i must be small. The main technical tool are latticesize independent estimates of LasotaYorke type which show that the transfer (PerronFrobenius) operator of the coupled system is quasicompact as an operator on the space of functions of bounded variation. 1 Introduction The purpose of this note is to summarize results from [11] and from the unpublished thesis [12]. Let L be a finite or countable index set, e.g. L = Zor L = Zn dZ. We investigate timediscrete dynamics on the state space X = [0; 1] L that are composed of independent chaotic actions on each component [0; 1] of X followed by some weak interaction that does not destroy the chaotic character of the whole system. More specifically, let ø : [0; 1] ! [0; 1]...},
author = {Keller, Gerhard},
booktitle = {Stochastic and spatial structures of dynamical systems (Amsterdam, 1995)},
editor = {S.J. van Strien, S.M. Verduyn Lunel},
faupublication = {yes},
keywords = {bounded variation map lattice via transfer operator lasotayorke type zor zn dz main technical tool coupled system general tent map chaotic character whole system state space markov partition timediscrete dynamic weak interaction latticesize independent estimate countable index set transfer operator approach total influence local map unpublished thesis map lattice independent chaotic action},
pages = {7180},
peerreviewed = {unknown},
publisher = {NorthHolland, Amsterdam},
series = {Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45},
title = {{Coupled} map lattice via transfer operators on functions of bounded variation},
year = {1996}
}
@article{faucris.320531059,
abstract = {We derive a dimensionreduction limit for a threedimensional rod with material voids by means of Γconvergence. Hereby, we generalize the results of the purely elastic setting [M. G. Mora and S. Müller, Derivation of the nonlinear bendingtorsion theory for inextensible rods by Γconvergence, Calc. Var. Partial Differential Equations 18 (2003) 287305] to a framework of free discontinuity problems. The effective onedimensional model features a classical elastic bendingtorsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a discontinuity of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [M. Friedrich, L. Kreutz and K. Zemas, Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces, preprint (2021), arXiv:2107.10808]. },
author = {Friedrich, Manuel and Kreutz, Leonard and Zemas, Konstantinos},
doi = {10.1142/S0218202524500131},
faupublication = {yes},
journal = {Mathematical Models & Methods in Applied Sciences},
keywords = {curvature regularization; dimension reduction; fracture; Geometric rigidity; rod theories; variable domains},
note = {CRISTeam Scopus Importer:20240405},
pages = {723777},
peerreviewed = {Yes},
title = {{Derivation} of effective theories for thin {3D} nonlinearly elastic rods with voids},
volume = {34},
year = {2024}
}
@article{faucris.202484603,
abstract = {Let be an infinite subset of {1,2,…}. We characterize
arithmetic and dynamical properties of the free set through group theoretical, topological and measure theoretic
properties of a set W (called the window) associated with . This
point of view stems from the interpretation of the set as a weak model set. Our main results are: is taut if and only
if the window is Haar regular; the dynamical system associated to is a Toeplitz system if and only if the window is topologically
regular; the dynamical system associated to is
proximal if and only if the window has empty interior; and the dynamical system
associated to has the "na"ively expected" maximal
equicontinuous factor if and only if the interior of the window is aperiodic.},
author = {Kasjan, Stanislaw and Keller, Gerhard and Lemańczyk, Mariusz},
doi = {10.1093/imrn/rnx196},
faupublication = {yes},
journal = {International Mathematics Research Notices},
pages = {26902734},
peerreviewed = {Yes},
title = {{Dynamics} of {B}{Free} sets: a view through the window},
url = {https://arxiv.org/abs/1702.02375},
volume = {2019},
year = {2019}
}
@article{faucris.122722424,
abstract = {Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such sets are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map which associates lattice subsets to points of the torus and then transferring the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so called weak model sets. In particular we prove pure point dynamical spectrum for the hull of a weak model set together with the push forward of the torus Haar measure under the torus parametrisation map, and we derive a formula for the pattern frequencies of configurations with maximal densit},
author = {Keller, Gerhard and Richard, Christoph},
doi = {10.1017/etds.2016.53},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {138},
peerreviewed = {Yes},
title = {{Dynamics} on the graph of the torus parametrisation},
volume = {38},
year = {2016}
}
@article{faucris.116154764,
abstract = {We construct a realanalytic circle map for which the corresponding PerronFrobenius operator has a realanalytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon C^{1}functions.},
author = {Keller, Gerhard and Rugh, Hans Henrik},
doi = {10.1088/09517715/17/5/009},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2004.nat.dma.pma29.eigenf},
pages = {17231730},
peerreviewed = {Yes},
title = {{Eigenfunctions} for smooth expanding circle maps},
volume = {17},
year = {2004}
}
@book{faucris.124122504,
abstract = {The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of methods and differing conventions.
The aim of this book is to identify the unifying threads by providing surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. Two major threads, emphasized throughout the book, are variational principles and Ljapunov functionals. The book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. Indepth surveys covering the macroscopic, microscopic and probabilistic approaches follow. Part I gives a basic introduction from the views of thermodynamics and probability theory. Part II collects surveys that look at the macroscopic approach of continuum mechanics and physics. Part III deals with the microscopic approach exposing the role of entropy as a concept in probability theory, namely in the analysis of the large time behavior of stochastic processes and in the study of qualitative properties of models in statistical physics. Finally in Part IV applications in dynamical systems, ergodic and information theory are presented.
The chapters were written to provide as cohesive an account as possible, making the book accessible to a wide range of graduate students and researchers. Any scientist dealing with systems that exhibit entropy will find the book an invaluable aid to their understandin},
address = {Princeton},
author = {Warnecke, Gerald and Keller, Gerhard and Greven, Andreas},
faupublication = {yes},
isbn = {9780691113388},
note = {UnivISImport:20150402:Pub.2003.nat.dma.pma29.entrop},
peerreviewed = {unknown},
publisher = {Princeton University Press},
title = {{Entropy}},
year = {2003}
}
@article{faucris.120103984,
abstract = {For piecewise monotone interval maps, we show that the KolmogorovSinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.},
author = {Brandt, Christoph and Keller, Gerhard and Pompe, Bernd},
doi = {10.1088/09517715/15/5/312},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2002.nat.dma.pma29.entrop},
pages = {15951602},
peerreviewed = {Yes},
title = {{Entropy} of interval maps via permutations},
volume = {15},
year = {2002}
}
@article{faucris.117417784,
abstract = {For a class of piecewise monotone interval maps T (including unimodal maps with negative Schwarzian derivative) and real valued functions f of bounded variation we compare equilibrium states μ of f with Hausdorff measures v and give an integral test for the dichotomy μ ≪ v or μ ⊥ v. For certain classes of rational maps such a result was proved in [15] and [3].},
author = {Hofbauer, Franz and Keller, Gerhard},
doi = {10.1002/mana.19931640117},
faupublication = {yes},
journal = {Mathematische Nachrichten},
pages = {239257},
peerreviewed = {Yes},
title = {{Equilibrium} states and {Hausdorff} measures for interval maps},
volume = {164},
year = {1993}
}
@article{faucris.122084424,
author = {Hofbauer, Franz and Keller, Gerhard},
doi = {10.1017/S014338570000955X},
faupublication = {no},
journal = {Ergodic Theory and Dynamical Systems},
pages = {2343},
peerreviewed = {Yes},
title = {{Equilibrium} states for piecewise monotonic transformations},
volume = {2},
year = {1982}
}
@article{faucris.121017204,
abstract = {For Sunimodal maps f , we study equilibrium states maximizing the free energies F t () := h() t R log jf 0 jd and the pressure function P (t) := sup F t (). It is shown that if f is uniformly hyperbolic on periodic orbits, then P (t) is analytic for t 1. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for t = 1 and of observable measure do not coincide.},
author = {Bruin, Hendrik Pieter and Keller, Gerhard},
doi = {10.1017/S0143385798108337},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
keywords = {equilibrium state sunimodal map free energy uniformly hyperbolic periodic orbit log jf pressure function observable measure},
pages = {765789},
peerreviewed = {Yes},
title = {{Equilibrium} states for {S}unimodal maps},
volume = {18},
year = {1998}
}
@book{faucris.118703244,
abstract = {This book provides a detailed introduction to the ergodic theory of equilibrium states giving equal weight to two of its most important applications, namely to equilibrium statistical mechanics on lattices and to (time discrete) dynamical systems. It starts with a chapter on equilibrium states on finite probability spaces that introduces the main examples for the theory on an elementary level. After two chapters on abstract ergodic theory and entropy, equilibrium states and variational principles on compact metric spaces are introduced, emphasizing their convex geometric interpretation. Stationary Gibbs measures, large deviations, the Ising model with external field, Markov measures, SinaiBowenRuelle measures for interval maps and dimension maximal measures for iterated function systems are the topics to which the general theory is applied in the last part of the book. The text is self contained except for some measure theoretic prerequisites that are listed (with references to the literature) in an appendix.},
address = {Cambridge},
author = {Keller, Gerhard},
faupublication = {yes},
isbn = {9780521595346},
note = {UnivISImport:20150402:Pub.1998.nat.dma.pma29.equili},
peerreviewed = {unknown},
publisher = {Cambridge University Press},
series = {London Mathematical Society Student Texts},
title = {{Equilibrium} {States} in {Ergodic} {Theory}},
volume = {42},
year = {1998}
}
@article{faucris.117444404,
author = {Keller, Gerhard},
faupublication = {no},
journal = {Comptes Rendus de l'Académie des Sciences  Série A: Mathématiques},
pages = {A625A627},
peerreviewed = {unknown},
title = {{Ergodicité} et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan},
volume = {289},
year = {1979}
}
@article{faucris.117443084,
author = {Hofbauer, Franz and Keller, Gerhard},
doi = {10.1007/BF01215004},
faupublication = {no},
journal = {Mathematische Zeitschrift},
pages = {119140},
peerreviewed = {Yes},
title = {{Ergodic} properties of invariant measures for piecewise monotonic transformations},
volume = {180},
year = {1982}
}
@article{faucris.285663451,
abstract = {We consider atomistic systems consisting of interacting particles arranged in atomic lattices whose quasistatic evolution is driven by timedependent boundary conditions. The interaction of the particles is modeled by classical interaction potentials where we implement a suitable irreversibility condition modeling the breaking of atomic bonding. This leads to a delay differential equation depending on the complete history of the deformation at previous times. We prove existence of solutions and provide numerical tests for the prediction of quasistatic crack growth in particle systems.},
author = {Badal, Rufat and Friedrich, Manuel and Seutter, Joscha},
doi = {10.1016/j.finmec.2022.100138},
faupublication = {yes},
journal = {Forces in Mechanics},
keywords = {Atomistic systems; Delay differential equations; Irreversibility condition; Minimizing movements; Quasistatic crack growth},
note = {CRISTeam Scopus Importer:20221125},
peerreviewed = {Yes},
title = {{Existence} of quasistatic crack evolution for atomistic systems},
volume = {9},
year = {2022}
}
@article{faucris.123050004,
abstract = {We prove exponential weak Bernoulli mixing for invariant measures of certain piecewise monotone interval maps studied in [BK] and [KN]. In particular we prove this for unimodal maps with negative Schwarzian derivative satisfying limliminfn→∞DTn(Tc)−−−−−−−−−√n>1, wherec is the unique critical point ofT.},
author = {Keller, Gerhard},
doi = {10.1007/BF02773683},
faupublication = {yes},
journal = {Israel Journal of Mathematics},
pages = {301310},
peerreviewed = {Yes},
title = {{Exponential} weak {Bernoulli} mixing for {Collet}{Eckmann} maps},
volume = {86},
year = {1994}
}
@article{faucris.123844204,
author = {Keller, Gerhard},
doi = {10.1017/S0143385700005861},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {717744},
peerreviewed = {Yes},
title = {{Exponents}, attractors and {Hopf} decompositions for interval maps},
volume = {10},
year = {1990}
}
@article{faucris.107777164,
abstract = {We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ (1 ℓ_{1}) where ℓ_{1} is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.},
author = {Keller, Gerhard and Nowicki, Tomasz},
doi = {10.1017/S0143385700008269},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {99120},
peerreviewed = {Yes},
title = {{Fibonacci} maps re(al)visited},
volume = {15},
year = {1995}
}
@article{faucris.120334984,
author = {van Hemmen, Jan L. and Keller, Gerhard and Kühn, Reimer},
doi = {10.1209/02955075/5/7/016},
faupublication = {no},
journal = {EPL  Europhysics Letters},
pages = {663668},
peerreviewed = {Yes},
title = {{Forgetful} memories},
volume = {5},
year = {1988}
}
@article{faucris.287257726,
abstract = {We study an atomistic model that describes the microscopic formation of material voids inside elastically stressed solids under an additional curvature regularization at the discrete level. Using a discretetocontinuum analysis, by means of a recent geometric rigidity result in variable domains (Friedrich et al 2021 arXiv:2107.10808) and Γconvergence tools, we rigorously derive effective linearized continuum models for elastically stressed solids with material voids in threedimensional elasticity.},
author = {Friedrich, Manuel and Kreutz, Leonard and Zemas, Konstantinos},
doi = {10.1088/13616544/aca5de},
faupublication = {yes},
journal = {Nonlinearity},
keywords = {atomistic systems; discretetocontinuum limits; free discontinuity problems; functions of bounded deformation; material voids; Γconvergence},
month = {Jan},
note = {CRISTeam Scopus Importer:20230106},
peerreviewed = {Yes},
title = {{From} atomistic systems to linearized continuum models for elastic materials with voids},
volume = {36},
year = {2023}
}
@article{faucris.110052624,
abstract = {We prove the quasicompactness of the PerronFrobenius operator of piecewise monotonic transformations when the inverse of the derivative is Höldercontinuous or, more generally, of bounded pvariation.},
author = {Keller, Gerhard},
doi = {10.1007/BF00532744},
faupublication = {no},
journal = {Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete},
pages = {461478},
peerreviewed = {unknown},
title = {{Generalized} bounded variation and applications to piecewise monotonic transformations},
volume = {69},
year = {1985}
}
@article{faucris.120586004,
abstract = {We show that the classic examples of quasiperiodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results also remain valid in more general classes of quasiperiodically forced systems. Further, we include an elementary proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the structure of the attractor in an example with twodimensional fibres also introduced by Grebogi et al. © 2006 IOP Publishing Ltd and London Mathematical Society.},
author = {Glendinning, Paul and Jäger, Tobias and Keller, Gerhard},
doi = {10.1088/09517715/19/9/001},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2006.nat.dma.pma29.howcha},
pages = {20052022},
peerreviewed = {Yes},
title = {{How} chaotic are strange nonchaotic attractors},
year = {2006}
}
@article{faucris.110775764,
abstract = {In this paper we give a computable criterion for a piecewiseexpanding interval map T to be mixing, which at the same time not only establishes explicit bounds on the spectral gap of the associated PerronFrobenius operator acting on the space of functions of bounded variation, but also establishes strict contraction rates for this operator. Of course such a result cannot be completely general, but our procedure covers a number of examples with infT′ > 2 and the bounds derived for them compare favorably with other estimates found in the literature.},
author = {Keller, Gerhard},
doi = {10.1142/S0218127499001267},
faupublication = {yes},
journal = {International Journal of Bifurcation and Chaos},
note = {UnivISImport:20150305:Pub.1999.nat.dma.pma29.interv},
pages = {17771784},
peerreviewed = {Yes},
title = {{Interval} maps with strictly contracting {Perron}{Frobenius} operators},
volume = {9},
year = {1999}
}
@article{faucris.205799527,
author = {Fadaei, Sara and Keller, Gerhard and Ghane, Fatemeh Helen},
doi = {10.1088/13616544/aae024},
faupublication = {yes},
journal = {Nonlinearity},
keywords = {invariant graph, skew product, synchronization, negative Schwarzian derivative, pinch points},
pages = {5329 },
peerreviewed = {Yes},
title = {{Invariant} graphs for chaotically driven maps},
volume = {31},
year = {2018}
}
@article{faucris.107995844,
abstract = {Simulation of plasma immersion ion implantation (PIII) is a necessary and valid tool to optimize the treatment homogeneity by adjusting parameters like voltage, plasma density, pulse rise time or pulse length. It can be shown that the ion mass mi and pulse rise time tr are not two independent parameters. Instead, for sufficiently long pulses, when the plasma sheath reaches the stationary state, the dose distributions obtained are characterized by the quotient tr/√mi. Therefore, for lower ion masses, shorter rise times must be used to obtain the same relative range distribution. The spatial homogeneity is not affected as the natural scaling length, the initial matrix sheath thickness x^{ini}, depends only on the voltage and ion density and is independent of the ion mass. In this report twodimensional particleincell (PIC) simulations for PIII treatment of trench structures with different tr/√mi ratios are presented, encompassing the range of mi = 4131 (corresponding to He^{+}Xe^{+}), at a pulse rise time of 0.5 μs and a pulse voltage of 45 kV. The implantation profiles are shifted more to the surface for lower masses, as their relative rise time is longer.},
author = {Keller, Gerhard and Mändl, S. and Rauschenbach, B. and Rüde, Ulrich},
doi = {10.1016/S02578972(00)010392},
faupublication = {yes},
journal = {Surface & Coatings Technology},
note = {UnivISImport:20150309:Pub.2001.tech.IMMD.lsinfs.ionmas},
pages = {117121},
peerreviewed = {Yes},
title = {{Ion} mass and scaling effects in {PIII} simulation},
url = {http://www.sciencedirect.com/science/article/pii/S0257897200010392/pdfft?md5=814e971b47a127467296cf78e78c1d27πd=1s2.0S0257897200010392main.pdf},
volume = {136},
year = {2001}
}
@article{faucris.281706343,
abstract = {Modifying Besicovitch’s construction of a set B of positive integers whose set of multiples MB has no asymptotic density, we provide examples of such sets B for which η:=1Z\MB∈{0,1}Z is a Toeplitz sequence. Moreover our construction produces examples, for which η is not only quasigeneric for the Mirsky measure (which has discrete dynamical spectrum), but also for some measure of positive entropy. On the other hand, modifying slightly an example from Kasjan, Keller, and Lemańczyk, we construct a set B for which η is an irregular Toeplitz sequence but for which the orbit closure of η in { 0 , 1 } ^{Z} is uniquely ergodic.},
author = {Keller, Gerhard},
doi = {10.1007/s00605022017546},
faupublication = {yes},
journal = {Monatshefte für Mathematik},
keywords = {Bfree dynamics; Density; Irregular Toeplitz sequence; Sets of multiples},
note = {CRISTeam Scopus Importer:20220916},
peerreviewed = {Yes},
title = {{Irregular} {B} free {Toeplitz} sequences via {Besicovitch}’s construction of sets of multiples without density},
year = {2022}
}
@article{faucris.122089484,
abstract = {Generalizing a theorem ofHofbauer (1979), we give conditions under which invariant measures for piecewise invertible dynamical systems can be lifted to Markov extensions. Using these results we prove:

(1)
IfT is anSunimodal map with an attracting invariant Cantor set, then ∫logT′dμ=0 for the unique invariant measure μ on the Cantor set.

(2)
IfT is piecewise invertible, iff is the RadonNikodym derivative ofT with respect to a σfinite measurem, if logf has bounded distortion underT, and if μ is an ergodicTinvariant measure satisfying a certain lower estimate for its entropy, then μ≪m iffh _{μ} (T)=Σlogf dμ.
},
author = {Keller, Gerhard},
doi = {10.1007/BF01308670},
faupublication = {yes},
journal = {Monatshefte für Mathematik},
pages = {183200},
peerreviewed = {Yes},
title = {{Lifting} measures to {Markov} extensions},
volume = {108},
year = {1989}
}
@article{faucris.111794144,
abstract = {We prove a local limit theorem for Lipschitz continuous observables on a weakly coupled lattice of piecewise expanding mixing interval maps. The core of the paper is a proof that the spectral radii of the Fouriertransfer operators for such a system are strictly less than 1. This extends the approach of [9] where the ordinary transfer operator was studied. © 2007 World Scientific Publishing Company.},
author = {Bardet, JeanBaptiste and Gouëzel, Sébastien and Keller, Gerhard},
doi = {10.1142/S0219493707001913},
faupublication = {yes},
journal = {Stochastics and Dynamics},
keywords = {Coupled map lattice; Local limit theorem; Piecewise expanding map; Spectral gap},
note = {UnivISImport:20150309:Pub.2007.nat.dma.pma29.limitt},
pages = {1736},
peerreviewed = {Yes},
title = {{Limit} theorems for coupled interval maps},
year = {2007}
}
@incollection{faucris.117447044,
author = {Denker, Manfred and Keller, Gerhard and Puri, Madan Lal},
booktitle = {New perspectives in theoretical and applied statistics (Bilbao, 1986)},
editor = {Madan Lal Puri, Jose Perez Vilaplana, Wolfgang Wertz},
faupublication = {no},
pages = {171206},
peerreviewed = {unknown},
publisher = {Wiley, New York},
series = {Wiley Series in Probability and Statistics},
title = {{Linear} rank statistics, bounded operators, and weak convergence},
year = {1987}
}
@incollection{faucris.117419544,
author = {Keller, Gerhard},
booktitle = {Lyapunov exponents (Oberwolfach, 1990)},
doi = {10.1007/BFb0086671},
editor = {Ludwig Arnold, Hans Crauel, JeanPierre Eckmann},
faupublication = {yes},
pages = {216226},
peerreviewed = {unknown},
publisher = {Springer, Berlin},
series = {Lecture Notes in Mathematics},
title = {{Lyapunov} exponents and complexity for interval maps},
volume = {1486},
year = {1991}
}
@article{faucris.123138884,
abstract = {We introduce a new coupled map lattice model in which the weak interaction takes place via rare “collisions”. By “collision” we mean a strong (possibly discontinuous) change in the system. For such models we prove uniqueness of the SRB measure and exponential spacetime decay of correlations.},
author = {Keller, Gerhard and Liverani, Carlangelo},
doi = {10.1007/s002200090835z},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {591597},
peerreviewed = {Yes},
title = {{Map} lattices coupled by collisions},
volume = {291},
year = {2009}
}
@article{faucris.117446604,
author = {Keller, Gerhard},
doi = {10.2307/2001395},
faupublication = {yes},
journal = {Transactions of the American Mathematical Society},
pages = {433497},
peerreviewed = {Yes},
title = {{Markov} extensions, zeta functions, and {Fredholm} theory for piecewise invertible dynamical systems},
volume = {314},
year = {1989}
}
@book{faucris.107800484,
abstract = {Was hat Mathematik mit den Lebenswissenschaften zu tun? Kann man die belebte Natur überhaupt in Formeln fassen? Und wenn ja, warum sollte man das? Naturwissenschaftliches Verständnis von Lebensvorgängen gewinnt man, indem man Theorien an Beobachtungen und Experimenten misst. In diesem Prozess spielt Mathematik sowohl bei der Theoriebildung – Stichwort: Modellierung – als auch bei der Überprüfung der Theorie an der Realität – Stichwort: Statistik – eine wichtige Rolle. Anders als in herkömmlichen Lehrbüchern bilden daher Modellbildung und Statistik den Kern dieses einführenden Buches. Viele Beispiele werden mit der freien Statistiksoftware R bearbeitet, und der Anhang bietet eine anwendungsorientierte Einführung in R durch „Learning by Doing“},
author = {Keller, Gerhard},
faupublication = {yes},
isbn = {9783825234935},
keywords = {Wachstums und Populationsmodelle • Grafische Darstellung von Daten und beschreibende Statistik; Modellierung mit Differenzialgleichungen; Wahrscheinlichkeitsrechnung; Testen und Schätzen; Korrelation und Regression; Sequence Alignment},
peerreviewed = {unknown},
publisher = {Ulmer GmbH},
title = {{Mathematik} in den {Life} {Sciences}},
year = {2011}
}
@article{faucris.123841344,
abstract = {It is shown that a finite system of coupled mixing tent maps has a unique absolutely continuous invariant measure and is exact with respect to this measure provided the coupling strength does not exceed a certain value ffl uni which is independent of the size of the system. 1 Introduction and main result Systems of coupled maps were widely studied during the last years, mostly by physicists, see [10, 5] for reviews. On the mathematical side two approaches to their understanding emerged: Initiated by Bunimovich and Sinai [3] several authors constructed models of statistical mechanics equivalent to infinite systems of coupled maps and studied invariant measures in this setting. Recent expositions of this type of approach are given in [2, 1]. On the other hand the dynamics of coupled maps were analyzed in terms of the transfer operator [8, 9], see also [7] for a brief review. In this note we continue this line of research. More specifically, we study finite systems of coupled tent maps ...},
author = {Keller, Gerhard},
faupublication = {yes},
journal = {Proceedings of the Steklov Institute of Mathematics},
keywords = {finite system coupled tent map coupled map main result system statistical mechanic coupled mixing tent map transfer operator coupling strength; invariant measure; several author; recent exposition; certain value; ffl; brief review; last year; mathematical side; continuous invariant measure},
note = {UnivISImport:20150305:Pub.1997.nat.dma.pma29.mixing},
pages = {315321},
peerreviewed = {Yes},
title = {{Mixing} for finite systems of coupled tent maps},
volume = {216},
year = {1997}
}
@article{faucris.117406784,
abstract = {Weak generalized synchrony in a driveresponse system occurs when the response dynamics is a unique but nondifferentiable function of the drive, in a manner that is similar to the formation of strange nonchaotic attractors in quasiperiodically driven dynamical systems. We consider a chaotically driven monotone map and examine the geometry of the limit set formed in the regime of weak generalized synchronization. The fractal dimension of the set of zeros is studied both analytically and numerically. We further examine the stable and unstable sets formed and measure the regularity of the coupling function. The stability index as well as the dimension spectrum of the equilibrium measure can be computed analytically. DOI: 10.1103/PhysRevE.87.042913},
author = {Keller, Gerhard and Jafri, Haider H. and Ramaswamy, R.},
doi = {10.1103/PhysRevE.87.042913},
faupublication = {yes},
journal = {Physical Review E},
peerreviewed = {Yes},
title = {{Nature} of weak generalized synchronization in chaotically driven maps},
volume = {87},
year = {2013}
}
@article{faucris.302880931,
abstract = {We characterize the passage from nonlinear to linearized Griffithfracture theories under noninterpenetration constraints. In particular, sequences of deformations satisfying a CiarletNečas condition in SBV^{2} and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD^{2} satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energyconvergent sequence of deformations fulfilling a CiarletNečas condition. The proof relies on a piecewise KornPoincaré inequality in GSBD^{2}, on a careful blowup analysis around jump points, as well as on a refined GSBD^{2}density result guaranteeing enhanced contact conditions for the approximants.},
author = {Almi, Stefano and Davoli, Elisa and Friedrich, Manuel},
doi = {10.1016/j.matpur.2023.05.001},
faupublication = {yes},
journal = {Journal De Mathematiques Pures Et Appliquees},
keywords = {CiarletNečas; Contact condition; Griffith fracture; Linearization; Noninterpenetration},
note = {CRISTeam Scopus Importer:20230526},
peerreviewed = {Yes},
title = {{Non}interpenetration conditions in the passage from nonlinear to linearized {Griffith} fracture},
year = {2023}
}
@article{faucris.323369758,
abstract = {We study the quasistatic evolution of a linear peridynamic KelvinVoigt viscoelastic material. More specifically, we consider the gradient flow of a nonlocal elastic energy with respect to a nonlocal viscous dissipation. Following an evolutionary Γconvergence approach, we prove that the solutions of the nonlocal problem converge to the solution of the local problem, when the peridynamic horizon tends to 0, that is, in the nonlocaltolocal limit.},
author = {Friedrich, Manuel and Seitz, Manuel and Stefanelli, Ulisse},
doi = {10.2478/caim20240001},
faupublication = {yes},
journal = {Communications in Applied and Industrial Mathematics},
keywords = {evolutionary Γconvergence; KelvinVoigt rheology; nonlocaltolocal limit; Peridynamics; viscoelasticity},
month = {Jan},
note = {CRISTeam Scopus Importer:20240607},
peerreviewed = {unknown},
title = {{Nonlocal}tolocal limit in linearized viscoelasticity},
volume = {15},
year = {2024}
}
@article{faucris.307851604,
abstract = {We derive an effective onedimensional limit from a threedimensional Kelvin–Voigt model for viscoelastic thinwalled beams, in which the elastic and the viscous stress tensor comply with a frameindifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Γ convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Γ convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).},
author = {Friedrich, Manuel and Machill, Lennart},
doi = {10.1007/s00526023025253},
faupublication = {yes},
journal = {Calculus of Variations and Partial Differential Equations},
note = {CRISTeam Scopus Importer:20230721},
peerreviewed = {Yes},
title = {{One}dimensional viscoelastic von {Kármán} theories derived from nonlinear thinwalled beams},
volume = {62},
year = {2023}
}
@article{faucris.124041984,
abstract = {We study the asymptotic behaviour of the solution of the stochastic difference equation Xn+1=Xn+g(Xn)(1+ξn+1), where g is a positive function, (ξn) is a 0mean, squareintegrable martingale difference sequence, and the states Xn<0 are assumed to be absorbing. We clarify, under which conditions Xn diverges with positive probability, satisfies a law of large numbers, and, properly normalized, converges in distribution. Controlled GaltonWatson processes furnish examples for the processes under consideration.},
author = {Keller, Gerhard and Kersting, Götz and Rösler, Uwe},
faupublication = {no},
journal = {Annals of Probability},
pages = {305343},
peerreviewed = {Yes},
title = {{On} the asymptotic behaviour of discrete time stochastic growth processes},
url = {http://links.jstor.org/sici?sici=00911798(198701)15:1<305:OTABOD>2.0.CO;2B&origin=MSN},
volume = {15},
year = {1987}
}
@article{faucris.117446824,
abstract = {We investigate the asymptotic behaviour of first passage times of diffusions or birthdeath processes. Necessary and sufficient conditions in terms of moments and the speed and scale functions are given for convergence of the first passage times to a normal or an exponential distribution.},
author = {Keller, Gerhard and Kersting, Götz and Rösler, Uwe},
doi = {10.1007/BF00319295},
faupublication = {no},
journal = {Probability Theory and Related Fields},
pages = {379395},
peerreviewed = {Yes},
title = {{On} the asymptotic behaviour of first passage times for diffusions},
volume = {77},
year = {1988}
}
@article{faucris.117448364,
abstract = {In this paper we study the asymptotic behaviour of the solution of the stochastic differential equation dX _{t}=g(X _{t})dt+σ(X _{t})dW _{t}, where σ and g are positive functions and W _{t}is a Wiener process. We clarify, under which conditions X _{t}may be approximated on {X _{t}→∞} by means of a deterministic function. Further the question is treated, whether X _{t}converges in distribution on {X _{t}→∞. We deal with the Itosolution as well as the Stratonovitchsolution and compare both.},
author = {Keller, Gerhard and Kersting, Götz and Rösler, Uwe},
doi = {10.1007/BF00531776},
faupublication = {no},
journal = {Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete},
pages = {163189},
peerreviewed = {unknown},
title = {{On} the asymptotic behaviour of solutions of stochastic differential equations},
volume = {68},
year = {1984}
}
@article{faucris.106586084,
author = {Keller, Gerhard},
doi = {10.2307/2047858},
faupublication = {yes},
journal = {Proceedings of the American Mathematical Society},
pages = {5153},
peerreviewed = {Yes},
title = {{On} the clustering conjecture for {Bernoulli} factors of {Bernoulli} shifts},
volume = {111},
year = {1991}
}
@article{faucris.111276044,
abstract = {We examine the behaviour of typical orbits in the Harper map, a quasiperiodically driven skewproduct dynamical system in two dimensions. When the map is critical, namely when all Lyapunov exponents vanish, the dynamics is parabolic.},
author = {Datta, Sandip and Jäger, Tobias and Keller, Gerhard and Ramaswamy, R.},
doi = {10.1088/09517715/17/6/017},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2004.nat.dma.pma29.onthed},
pages = {23152323},
peerreviewed = {Yes},
title = {{On} the dynamics of the critical {Harper} map},
volume = {17},
year = {2004}
}
@article{faucris.288293011,
abstract = {We prove that on Bfree subshifts, with B satisfying the Erds condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.},
author = {Keller, Gerhard and Lemanczyk, Mariusz and Richard, Christoph and Sell, Daniel},
doi = {10.1007/s1185602224379},
faupublication = {yes},
journal = {Israel Journal of Mathematics},
note = {CRISTeam WoS Importer:20230127},
pages = {567594},
peerreviewed = {Yes},
title = {{On} the {Garden} of {Eden} theorem for {B}free subshifts},
volume = {251},
year = {2022}
}
@article{faucris.117447704,
abstract = {We determine the essential spectral radius of the PerronFrobeniusoperator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such onedimensional systems to the greatest eigenvalues of generalized PerronFrobeniusoperators of the transformations (operators which yield singular invariant measures).},
author = {Keller, Gerhard},
doi = {10.1007/BF01240219},
faupublication = {no},
journal = {Communications in Mathematical Physics},
pages = {181193},
peerreviewed = {Yes},
title = {{On} the rate of convergence to equilibrium in onedimensional systems},
url = {http://projecteuclid.org/euclid.cmp/1103941781},
volume = {96},
year = {1984}
}
@article{faucris.117445504,
author = {Denker, Manfred and Keller, Gerhard and Urbański, Mariusz},
faupublication = {yes},
journal = {Studia Mathematica},
pages = {2736},
peerreviewed = {Yes},
title = {{On} the uniqueness of equilibrium states for piecewise monotone mappings},
volume = {97},
year = {1990}
}
@article{faucris.117439564,
author = {Denker, Manfred and Keller, Gerhard},
doi = {10.1007/BF00534953},
faupublication = {no},
journal = {Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete},
pages = {505522},
peerreviewed = {unknown},
title = {{On} {U}statistics and v. {Mises}' statistics for weakly dependent processes},
volume = {64},
year = {1983}
}
@incollection{faucris.122795464,
author = {Keller, Gerhard},
booktitle = {Stochastic modelling in biology (Heidelberg, 1988)},
editor = {Petre Tautu},
faupublication = {yes},
pages = {412419},
peerreviewed = {unknown},
publisher = {World Sci. Publ., Teaneck, NJ},
title = {{Periodic} orbits for interval maps},
year = {1990}
}
@article{faucris.202490735,
abstract = {There is a renewed interest in weak model sets due to their connection to
free systems, which emerged from Sarnak's program on the M"obius
disjointness conjecture. Here we continue our recent investigation
[arXiv:1511.06137] of the extended hull GW, a dynamical system naturally associated to a weak
model set in an abelian group G with relatively compact window W. For
windows having a nowhere dense boundary (this includes compact windows), we
identify the maximal equicontinuous factor of GW and give a sufficient condition when GW is an almost 1:1 extension of
its maximal equicontinuous factor. If the window is measurable with positive
Haar measure and is almost compact, then the system GW equipped with its Mirsky
measure is isomorphic to its Kronecker factor. For general nontrivial ergodic
probability measures on GW, we provide a kind of lower bound for the Kronecker
factor. All relevant factor systems are natural Gactions on quotient
subgroups of the torus underlying the weak model set. These are obtained by
factoring out suitable window periods. Our results are specialised to the usual
hull of the weak model set, and they are also interpreted for free systems.},
author = {Keller, Gerhard and Richard, Christoph},
doi = {10.1007/s1185601817888},
faupublication = {yes},
journal = {Israel Journal of Mathematics},
pages = {85132},
peerreviewed = {Yes},
title = {{Periods} and factors of weak model sets},
volume = {229},
year = {2019}
}
@article{faucris.111694044,
abstract = {We construct a mixing continuous piecewise linear map on [1, 1] with the property that a twodimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borrowed from Gielis and MacKay (2000 Nonlinearity 13 86788); namely, we compare the dynamics of the CML with those of a probabilistic cellular automaton of Toom's type; see MacKay (2005 Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems (Lecture Notes in Physics vol 671) ed JR Chazottes and B Fernandez (Berlin: Springer) pp 6594) for a detailed discussion. © 2006 IOP Publishing Ltd and London Mathematical Society.},
author = {Bardet, JeanBaptiste and Keller, Gerhard},
doi = {10.1088/09517715/19/9/012},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2006.nat.dma.pma29.phaset},
pages = {21932210},
peerreviewed = {Yes},
title = {{Phase} transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling},
year = {2006}
}
@incollection{faucris.119791584,
author = {Keller, Gerhard},
booktitle = {Seminar on Probability, Rennes 1978 (French)},
faupublication = {no},
pages = {Exp. No. 6, 32},
peerreviewed = {unknown},
publisher = {Univ. Rennes, Rennes},
title = {{Piecewise} monotonic transformations and exactness},
year = {1978}
}
@article{faucris.122182984,
abstract = {We discuss the statistical error in the calculation of the sample correlation integral from a finite sample of points. For this purpose we introduce an estimator of the covariance matrix of these estimators. The application of the method is described and it is shown that only small modifications to a standard GrassbergerProcaccia algorithm are necessary. Testing the method with 100 independent runs of the Hénon system, we show that the errors obtained for the correlation integrals are in good accordance with the sample error. These results are extended to the application to timecontinuous systems, in our case the Lorenz system},
author = {Frank, M. and Keller, Gerhard and Sporer, Ralph},
doi = {10.1103/PhysRevE.53.5831},
faupublication = {yes},
journal = {Physical Review E  Statistical, Nonlinear, and Soft Matter Physics},
pages = {58315836},
peerreviewed = {unknown},
title = {{Practical} implementation of error estimation for the correlation dimension},
volume = {6},
year = {1996}
}
@incollection{faucris.117413824,
abstract = {Let (f _{ t })o≤t≤1 denote the family of quadratic maps f _{ t(x) } = 2t(1 x ^{2})  1 on [1, 1]. An important aspect of the asymptotics of interates of a map f _{ t } is the behaviour of mass distributions along individual orbits.},
author = {Hofbauer, Franz and Keller, Gerhard},
booktitle = {Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992)},
doi = {10.1007/9781461303213{\_}7},
editor = {Y. Takahashi},
faupublication = {yes},
pages = {8994},
peerreviewed = {unknown},
publisher = {Plenum, New York},
title = {{Quadratic} maps with maximal oscillation},
year = {1995}
}
@article{faucris.121567424,
abstract = {An interval map is said to have an asymptotic measure if the time averages of the iterates of Lebesgue measure converge weakly. We construct quadratic maps which have no asymptotic measure. We also find examples of quadratic maps which have an asymptotic measure with very unexpected properties, e.g. a map with the point mass on an unstable fix point as asymptotic measure. The key to our construction is a new characterization of kneading sequences.},
author = {Hofbauer, Franz and Keller, Gerhard},
doi = {10.1007/BF02096761},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {319337},
peerreviewed = {Yes},
title = {{Quadratic} maps without asymptotic measure},
url = {http://projecteuclid.org/euclid.cmp/1104180141},
volume = {127},
year = {1990}
}
@article{faucris.122121384,
abstract = {We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts of random topological dynamics.},
author = {Keller, Gerhard and Jäger, Tobias},
doi = {10.1090/tran/6591},
faupublication = {yes},
journal = {Transactions of the American Mathematical Society},
pages = {66436662},
peerreviewed = {Yes},
title = {{Random} minimality and continuity of invariant graphs in random dynamical systems},
volume = {368},
year = {2012}
}
@article{faucris.110646844,
abstract = {For piecewise expanding onedimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here 'eigenvalue' means eigenvalue of the corresponding PerronFrobenius operator acting on the space of functions of bounded variation.) This result applies e.g. to the approximation of the system by a finite state Markov chain and generalizes Ulam's conjecture about the approximation of the SinaiBowenRuelle invariant measure of such a map. We provide several simple examples showing that for maps with periodic turning points and for general multidimensional smooth hyperbolic maps isolated eigenvalues are typically unstable under random perturbations Our main tool in the onedimendional case is a special technique for 'interchanging' the map and the perturbation, developed in our previous paper (Blank M L and Keller G 1997 Stochastic stability versus localization in chaotic dynamical systems Nonlinearity 10 81107), combined with a compactness argument.},
author = {Blank, Michael and Keller, Gerhard},
doi = {10.1088/09517715/11/5/010},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150305:Pub.1998.nat.dma.pma29.random},
pages = {13511364},
peerreviewed = {Yes},
title = {{Random} perturbations of chaotic dynamical systems: {Stability} of the spectrum},
volume = {11},
year = {1998}
}
@article{faucris.120954944,
abstract = {We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on ℤ^{d}. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.},
author = {Dolgopyat, Dmitry and Keller, Gerhard and Liverani, Carlangelo},
doi = {10.1214/07AOP369},
faupublication = {yes},
journal = {Annals of Probability},
note = {UnivISImport:20150402:Pub.2007.nat.dma.pma29.random},
pages = {16761710},
peerreviewed = {Yes},
title = {{Random} walk in {Markovian} environment},
volume = {36},
year = {2007}
}
@article{faucris.117404364,
abstract = {We discuss how an eigenvalue perturbation formula for transfer operators of dynamical systems is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems. We also list a number of piecewise expanding systems to which this general theory applies and discuss the prospects to apply this theory to some classes of piecewise hyperbolic systems.},
author = {Keller, Gerhard and Liverani, Carlangelo},
doi = {10.1007/s1095500997478},
faupublication = {yes},
journal = {Journal of Statistical Physics},
pages = {519534},
peerreviewed = {Yes},
title = {{Rare} events, escape rates and quasistationarity: some exact formulae},
volume = {135},
year = {2009}
}
@article{faucris.123138004,
abstract = {We discuss how an eigenvalue perturbation formula for transfer operators of dynamical systems is related to exponential hitting time distributions and extreme value theory for processes generated by chaotic dynamical systems. We also list a number of piecewise expanding systems to which this general theory applies and discuss the prospects to apply this theory to some classes of piecewise hyperbolic systems.},
author = {Keller, Gerhard},
doi = {10.1080/14689367.2011.653329},
faupublication = {yes},
journal = {Dynamical SystemsAn International Journal},
pages = {1127},
peerreviewed = {Yes},
title = {{Rare} events, exponential hitting times and extremal indices via spectral perturbation},
volume = {27},
year = {2012}
}
@incollection{faucris.117412944,
abstract = {We discuss some statistical theory of the simultaneous estimation of correlation integrals from dynamical data with varying radii and embedding dimensions. Thereby we focus on the estimation of the covariance matrix of these estimators taking into account the finite sample size and the correlation time effects observed by Theiler [23]. As applications we discuss linear model statistics like linear regression estimates of correlation dimension and entropy and the detection of noise. 1 Introduction Let X 1 ; X 2 ; X 3 ; : : : be a realvalued stationary time series that is mixing in a sense to be made precise later. Typical examples would be a) independent identically distributed (i.i.d.) random observations, b) observation on a "chaotic" dynamical system, c) observations on a noisy system. In particular there may be some interesting dependence between consecutive observations that can be studied by looking at the distribution ¯ ` of blocks Y ` i := (X i ; : : : ; X i+`\Gamma1 ) 2 R ...},
author = {Keller, Gerhard and Sporer, Ralph},
booktitle = {Stochastic and spatial structures of dynamical systems (Amsterdam, 1995)},
editor = {S. J. van Strien, S. M. Verduyn Lunel},
faupublication = {yes},
pages = {1727},
peerreviewed = {unknown},
publisher = {NorthHolland, Amsterdam},
series = {Konink. Nederl. Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45},
title = {{Remarks} on the linear regression approach to dimension estimation},
volume = {45},
year = {1996}
}
@article{faucris.110048444,
abstract = {Various questions about the invariant measures of a dynamical system can be answered by computations of regular functionals or by ranking methods based on a set of observations. This includes symmetry tests and the determination of dimension coefficients. The paper contains the theoretical results and several simulations explain the methods.},
author = {Denker, Manfred and Keller, Gerhard},
doi = {10.1007/BF01010905},
faupublication = {no},
journal = {Journal of Statistical Physics},
keywords = {Dynamical systems; invariant measure; Hausdorff dimension; symmetric densities; U statistics; rank statistics},
pages = {6793},
peerreviewed = {Yes},
title = {{Rigorous} statistical procedures for data from dynamical systems},
volume = {44},
year = {1986}
}
@article{faucris.121554664,
abstract = {We extend a number of results from onedimensional dynamics based on spectral properties of the RuellePerronFrobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of a SinaiRuelleBowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulamtype finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finitedimensional problem.},
author = {Blank, Michael and Keller, Gerhard and Liverani, Carlangelo},
doi = {10.1088/09517715/15/6/309},
faupublication = {yes},
journal = {Nonlinearity},
note = {UnivISImport:20150309:Pub.2002.nat.dma.pma29.ruelle},
pages = {19051973},
peerreviewed = {Yes},
title = {{Ruelle}{Perron}{Frobenius} spectrum for {Anosov} maps},
volume = {15},
year = {2002}
}
@article{faucris.119999044,
abstract = {We study finite coupled map lattices of size d ⩾ 2 with individual maps τ: [0, 1] → [0, 1] and constant diffuse coupling. For τ (x) = 2x mod 1 we give sufficient conditions that the coupled system has a continuum of ergodic components. In the case d = 2 we determine the number of ergodic components for all coupling strengths. If τ is a mixing tent map close to the transition from mixing to a periodic interval with period 2, the uncoupled system is mixing, whereas numerical studies suggest that coupling with a suitable strength breaks up the phase space into domains which are interchanged with period 2. In case d = 2 we prove this rigorously.},
author = {Keller, Gerhard and Künzle, Martin and Nowicki, Tomasz},
doi = {10.1016/01672789(92)902052},
faupublication = {yes},
journal = {Physica DNonlinear Phenomena},
pages = {3951},
peerreviewed = {Yes},
title = {{Some} phase transitions in coupled map lattices},
volume = {59},
year = {1992}
}
@article{faucris.117445064,
author = {Hofbauer, Franz and Keller, Gerhard},
faupublication = {yes},
journal = {Annales de l'Institut Henri Poincaré  Physique Théorique},
pages = {413425},
peerreviewed = {Yes},
title = {{Some} remarks on recent results about {S}unimodal maps},
url = {http://www.numdam.org/item?id=AIHPA{\_}1990{\_}{\_}53{\_}4{\_}413{\_}0},
volume = {53},
year = {1990}
}
@article{faucris.117418004,
abstract = {We study unimodal interval mapsT with negative Schwarzian derivative satisfying the ColletEckmann condition DT ^{ n }(Tc)≧Kλ _{c} ^{n} for some constantsK>0 and λ_{c}>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the formSn=Σn−1i=0f(Tix)
, and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.
This is achieved by proving quasicompactness of the Perron Frobenius operator and of similar transfer operators for the Markov extension ofT and relating the isolated eigenvalues of these operators to the poles of the corresponding Ruelle zeta functions.},
author = {Keller, Gerhard and Nowicki, Tomasz},
doi = {10.1007/BF02096623},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {3169},
peerreviewed = {unknown},
title = {{Spectral} theory, zeta functions and the distribution of periodic points for {Collet}{Eckmann} maps},
url = {http://projecteuclid.org/euclid.cmp/1104251138},
volume = {149},
year = {1992}
}
@article{faucris.280965994,
abstract = {Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measuretheoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.},
author = {Keller, Gerhard and Richard, Christoph and Strungaru, Nicolae},
doi = {10.4153/S0008439522000352},
faupublication = {yes},
journal = {Canadian Mathematical BulletinBulletin Canadien De Mathematiques},
note = {CRISTeam WoS Importer:20220826},
peerreviewed = {Yes},
title = {{Spectrum} of weak model sets with {Borel} windows},
year = {2022}
}
@article{faucris.117400624,
author = {Keller, Gerhard},
doi = {10.1112/jlms/jdt070},
faupublication = {yes},
journal = {Journal of the London Mathematical SocietySecond Series},
pages = {603622},
peerreviewed = {Yes},
title = {{Stability} index for chaotically driven concave maps},
volume = {89},
year = {2014}
}
@article{faucris.112456784,
abstract = {Skew product systems with monotone onedimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov ma},
author = {Keller, Gerhard},
doi = {10.3934/dcdss.2017015},
faupublication = {yes},
journal = {AIMS Journal},
pages = {313334},
peerreviewed = {Yes},
title = {{Stability} index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors},
volume = {10},
year = {2017}
}
@article{faucris.110778624,
author = {Keller, Gerhard and Liverani, Carlangelo},
faupublication = {yes},
journal = {Annali della Scuola Normale Superiore di PisaClasse di Scienze},
note = {UnivISImport:20150305:Pub.1999.nat.dma.pma29.stabil},
pages = {141152},
peerreviewed = {Yes},
title = {{Stability} of the spectrum for transfer operators},
volume = {28},
year = {1999}
}
@article{faucris.266504073,
abstract = {Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two and threebody contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbours or angles formed by pairs of first neighbours. Such configurations are called anglerigid. We tackle this question in the class of finite configurations in Z(2), seen as planar threedimensional point sets. A sufficient condition preventing anglerigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.},
author = {Betermin, Laurent and Friedrich, Manuel and Stefanelli, Ulisse},
doi = {10.1088/13616544/ac3383},
faupublication = {yes},
journal = {Nonlinearity},
note = {CRISTeam WoS Importer:20211126},
pages = {83928413},
peerreviewed = {Yes},
title = {{Stability} of {Z}(2) configurations in {3D}},
volume = {34},
year = {2021}
}
@article{faucris.122625404,
abstract = {A Markov process on a compact metric space,X is given by random transformations.S is a finite set of continuous transformations ofX to itself. A random evolution onX is defined by lettingx inX evolve toT(x) forT inS with probability that depends onx andT but is independent of any other past measurable events. This type of model is often called a place dependent iterated function system. The transformations are assumed to have either monotone or contractive properties. Theorems are given to describe the number and types of ergodic invariant measures. Special emphasis is given to learning models and selfreinforcing random walks.},
author = {Burton, Robert M. and Keller, Gerhard},
doi = {10.1007/BF01046765},
faupublication = {yes},
journal = {Journal of Theoretical Probability},
keywords = {Random transformation; iterated function system; weak Bernoulli; selfreinforcing random walk},
pages = {116},
peerreviewed = {Yes},
title = {{Stationary} measures for randomly chosen maps},
volume = {6},
year = {1993}
}
@article{faucris.117402824,
abstract = {We study systems of globally coupled interval maps, where the identical individual maps have two expanding, fractional linear, onto branches, and where the coupling is introduced via a parameter  common to all individual maps  that depends in an analytic way on the mean field of the system. We show: 1) For the range of coupling parameters we consider, finitesize coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finitesize systems exhibit exponential decay of correlations. 2) For the same range of parameters, the selfconsistent PerronFrobenius operator which captures essential aspects of the corresponding infinitesize system (arising as the limit of the above when the system size tends to infinity), undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium.},
author = {Bardet, JeanBaptiste and Keller, Gerhard and Zweimüller, Roland},
doi = {10.1007/s0022000908549},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {237270},
peerreviewed = {Yes},
title = {{Stochastically} stable globally coupled maps with bistable thermodynamic limit},
volume = {292},
year = {2009}
}
@inproceedings{faucris.123581304,
address = {Berlin, Heidelberg},
author = {Keller, Gerhard},
booktitle = {Dynamical System and Chaos, Proceedings of the Sitges Conference on Statistical Mechanics Sitges},
date = {19820905/19820911},
doi = {10.1007/3540122761{\_}15},
faupublication = {no},
pages = {192193},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Stochastic} perturbations of some strange attractors},
venue = {Barcelona},
year = {1982}
}
@article{faucris.117441764,
abstract = {For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the PerronFrobeniusoperator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of PerronFrobeniusoperators.},
author = {Keller, Gerhard},
doi = {10.1007/BF01667385},
faupublication = {no},
journal = {Monatshefte für Mathematik},
pages = {313333},
peerreviewed = {Yes},
title = {{Stochastic} stability in some chaotic dynamical systems},
volume = {94},
year = {1982}
}
@inproceedings{faucris.106584764,
author = {Keller, Gerhard},
booktitle = {Ergodic Theory and Related Topics. Proceedings of the Conference held in Vitte/Hiddensee},
date = {19811019/19811023},
faupublication = {no},
pages = {123127},
peerreviewed = {Yes},
publisher = {Horst Michel, Akademischer Verlage},
series = {Math. Res.},
title = {{Stochastic} stability of some onedimensional dynamical systems},
venue = {Vitte  Hiddensee},
volume = {94},
year = {1982}
}
@article{faucris.123707144,
abstract = {We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which the localization phenomenon (i.e. stabilization of singular invariant measures) takes place are also considered. Our main tools are the socalled bounded variation approach combined with the ergodic theorem of Ionescu  Tulcea and Marinescu, and a random walk argument that we apply to prove the absence of `traps' under the action of random perturbations.},
author = {Blank, Michael and Keller, Gerhard},
doi = {10.1088/09517715/10/1/006},
faupublication = {yes},
journal = {Nonlinearity},
pages = {81107},
peerreviewed = {Yes},
title = {{Stochastic} stability versus localization in onedimensional chaotic dynamical systems},
volume = {10},
year = {1997}
}
@article{faucris.202487101,
abstract = {We study a class of globally coupled maps in the continuum limit,
where the individual maps are expanding maps of the circle. The circle
maps in question are such that the uncoupled system admits a unique
absolutely continuous invariant measure, which is furthermore mixing.
Interaction arises in the form of diffusive coupling, which involves a
function that is discontinuous on the circle. We show that for
sufficiently small coupling strength the coupled map system admits a
unique absolutely continuous invariant distribution, which depends on
the coupling strength ε. Furthermore, the invariant density
exponentially attracts all initial distributions considered in our
framework. We also show that the dependence of the invariant density on
the coupling strength ε is Lipschitz continuous in the BV norm.
When
the coupling is sufficiently strong, the limit behavior of the system
is more complex. We prove that a wide class of initial measures approach
a point mass with support moving chaotically on the circle. This can be
interpreted as synchronization in a chaoti},
author = {Bálint, Péter and Keller, Gerhard and Mincsovicsne Selley, Fanni and Tóth, Imre Péter},
doi = {10.1088/13616544/aac5b0},
faupublication = {yes},
journal = {Nonlinearity},
peerreviewed = {Yes},
title = {{Synchronization} versus stability of the invariant distribution for a class of globally coupled maps},
volume = {31},
year = {2018}
}
@article{faucris.202489588,
abstract = {For any set ⊆ℕ={1,2,…} one can define its
emph{set of multiples} :=⋃b∈bℤ and the set of emph{free numbers} :=ℤ∖. Tautness of the set
is a basic property related to questions around the asymptotic
density of ⊆ℤ. From a dynamical
systems point of view (originated by Sarnak) one studies η, the indicator
function of ⊆ℤ, its shiftorbit
closure Xη⊆{0,1}ℤ and the stationary probability
measure νη defined on Xη by the frequencies of finite blocks in
η. In this paper we prove that tautness implies the following two
properties of η: (1) The measure νη has full topological support
in Xη. (2) If Xη is proximal, i.e. if the onepoint set
{…000…} is contained in Xη and is the unique minimal subset
of Xη, then Xη is hereditary, i.e. if x∈Xη and if w is
an arbitrary element of {0,1}ℤ, then also the coordinatewise
product w⋅x belongs to Xη. This strengthens two results from
[Bartnicka et al. 2015] which need the stronger assumption that
has light tails for the same conclusions.},
author = {Keller, Gerhard},
doi = {10.4064/sm18030594},
faupublication = {yes},
journal = {Studia Mathematica},
pages = {205216},
peerreviewed = {Yes},
title = {{Tautness} of sets of multiples and applications to {B}free dynamics},
url = {https://arxiv.org/abs/1802.08309},
volume = {247},
year = {2019}
}
@article{faucris.123140204,
abstract = {We extend the sliding block code in symbolic dynamics to transform J(>= 2) sequences of Markov chains with time delays. Under the assumption that the chains are irreducible and aperiodic, we prove the central limit theorem (CLT) for the normalized sums of extended sliding block codes from J sequences of Markov chains. We apply the theorem to the system analysis of asynchronous spread spectrum multiple access (SSMA) communication systems using spreading sequences of Markov chains. We find that the standard Gaussian approximation (SGA) for estimations of bit error probabilities in such systems is the 0th order approximation of the evaluation based on the CLT. We also provide a simple theoretical evaluation of bit error probabilities in such systems, which agrees properly with the experimental results even for the systems with small number of users and low length of spreading sequences.},
author = {Fujisaki, Hiroshi and Keller, Gerhard},
doi = {10.1093/ietfec/e89a.9.2307},
faupublication = {yes},
journal = {IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences},
keywords = {sliding block code;Markov chain;a(N)mixing;central limit theorem (CLT);bit error probability;spread spectrum multiple access (SSMA) communication system},
pages = {23072314},
peerreviewed = {Yes},
title = {{The} central limit theorem for the normalized sums of the {MAI} for {SSMA} communication systems using spreading sequences of {Markov} chains},
volume = {E89A},
year = {2006}
}
@article{faucris.117404804,
author = {Keller, Gerhard and Jäger, Tobias},
doi = {10.1017/S0143385705000477},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {447465},
peerreviewed = {Yes},
title = {{The} {Denjoy} type of argument for quasiperiodically forced circle diffeomorphisms},
volume = {26},
year = {2006}
}
@article{faucris.285701747,
abstract = {In this paper we are interested in the possible values taken by the pair (lambda(1) (Omega); mu(1)(Omega)) the first eigenvalues of the Laplace operator with Dirichlet and Neumann boundary conditions respectively of a bounded plane domain Omega. We prove that, without any particular assumption on the class of open sets Omega, the two classical inequalities (the FaberKrahn inequality and the Weinberger inequality) provide a complete system of inequalities. Then we consider the case of convex plane domains for which we give new inequalities for the product lambda(1)mu(1). We plot the socalled BlaschkeSantalo diagram and give some conjectures.},
author = {Ftouhi, Ilias and Henrot, Antoine},
faupublication = {yes},
journal = {Mathematical Reports},
month = {Jan},
note = {CRISTeam WoS Importer:20221125},
pages = {159177},
peerreviewed = {Yes},
title = {{THE} {DIAGRAM} (lambda(1), mu(1))},
volume = {24},
year = {2022}
}
@article{faucris.273953583,
abstract = {Inspired by the modelization of 2D materials systems, we characterize arrangements of identical nonflat squares in 3D. We prove that the fine geometry of such arrangements is completely characterized in terms of patterns of mutual orientations of the squares and that these patterns are periodic and onedimensional. In contrast to the flat case, the nonflatness of the tiles gives rise to nontrivial geometries, with configurations bending, wrinkling, or even rolling up in one direction.},
author = {Friedrich, Manuel and Seitz, Manuel and Stefanelli, Ulisse},
doi = {10.1007/s00032022003505},
faupublication = {yes},
journal = {Milan Journal of Mathematics},
note = {CRISTeam WoS Importer:20220429},
peerreviewed = {Yes},
title = {{Tilings} with {Nonflat} {Squares}: {A} {Characterization}},
year = {2022}
}
@incollection{faucris.118916644,
abstract = {
We investigate the dynamics of Lorenz maps, in particular the asymptotical behaviour of the trajectory of typical points. For Lorenz maps f with negative Schwarzian derivative we give a classification of the possible metric attractors and show that either f has an ergodic absolutely continuous invariant probability measure of positive entropy or the iterates of typical points spend most of their time shadowing the trajectory of one of the two critical values. Our main tool therefore is the construction of Markov extensions for Lorenz maps which provide a unified framework to approach both the topological and the measurable aspects of the dynamics.
We study the bifurcation diagram of a smooth two parameter family of Lorenz maps which describes the parameter dependence of the kneading invariant and show that essentially every admissible kneading invariant actually occurs if the family is sufficiently rich. Finally, we adress the problem whether the kneading invariant depends monotonously on the parameters.
},
address = {New York u.a.},
author = {Keller, Gerhard and St. Pierre, Matthias},
booktitle = {Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems},
doi = {10.1007/9783642565892{\_}15},
editor = {Bernold Fiedler},
faupublication = {yes},
isbn = {9783642625244},
note = {UnivISImport:20150420:Pub.2001.nat.dma.pma29.topolo},
pages = {333361},
peerreviewed = {unknown},
publisher = {Springer},
title = {{Topological} and measurable dynamics of {Lorenz} maps},
year = {2001}
}
@article{faucris.117418444,
abstract = {Let L denote a finite or infinite onedimensional lattice. To each lattice site is attached a copy of a dynamical system with phase space [0, 1] and dynamics described by a transformation τ: [0, 1] → [0, 1], which is the same on each component. Denote the direct product of these identical systems by T: X → X where X = [0, 1]^{L}. From this product system we obtain a coupled map lattice (CML) S_{ε}: X → X, if we introduce some interaction between the components, e.g. by averaging between nearest neighbours. The strength of the coupling depends upon some parameter ε.
For a broad class of piecewise expanding singlecomponenttransformations τ we study such systems via their transfer operators and treat the coupled system as a perturbation of the uncoupled one. This yields existence and stability results for Tinvariant measures with absolutely continuous finitedimensional marginals.},
author = {Keller, Gerhard and Künzle, Martin},
doi = {10.1017/S0143385700006763},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
pages = {297318},
peerreviewed = {Yes},
title = {{Transfer} operators for coupled map lattices},
volume = {12},
year = {1992}
}
@article{faucris.117409644,
abstract = {
We study infinite systems of unidirectionally coupled interval maps on the lattice ^{d}. It is assumed that the singlesite map τ is piecewise C^{2} expanding, mixing and satisfies stably a LasotaYorketype condition, and that the interaction is such that each lattice site i is influenced only by sites from i + ^{d}. We concentrate on properties of timeinvariant probability measures whose finitedimensional conditional distributions are absolutely continuous w.r.t. the corresponding finitedimensional Lebesgue measure. For sufficiently weak interactions we prove:
• Any d≥1: Two measures from this class are different if and only if their restrictions to the spatial tail field are different. (The existence of such measures was established previously.)
• d = 1: If the interaction is superexponentially decreasing and if τ is continuous, then there is a unique measure in this class. It has exponentially decreasing correlations both in time and in space.
The key to these results is a probabilistic coupling procedure for nonautonomous dynamical systems.
},
author = {Keller, Gerhard and Zweimüller, Roland},
doi = {10.1088/09517715/15/1/301},
faupublication = {yes},
journal = {Nonlinearity},
pages = {124},
peerreviewed = {Yes},
title = {{Unidirectionally} coupled interval maps: between dynamics and statistical mechanics},
volume = {15},
year = {2002}
}
@article{faucris.109948564,
abstract = {We prove the existence of a unique SRB measure for a wide range of multidimensional weakly coupled map lattices. These include piecewise expanding maps with diffusive coupling.},
author = {Keller, Gerhard and Liverani, Carlangelo},
doi = {10.1007/s0022000514747},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {3350},
peerreviewed = {Yes},
title = {{Uniqueness} of the {SRB} measure for piecewise expanding weakly coupled map lattices in any dimension},
volume = {262},
year = {2006}
}
@article{faucris.117444184,
author = {Keller, Gerhard},
faupublication = {no},
journal = {Comptes Rendus de l'Académie des Sciences  Série A: Mathématiques},
pages = {A155A158},
peerreviewed = {unknown},
title = {{Un} théorème de la limite centrale pour une classe de transformations monotones par morceaux},
volume = {291},
year = {1980}
}
@article{faucris.117413164,
author = {Bruin, Hendrik Pieter and Keller, Gerhard and Nowicki, Tomasz and van Strien, Sebastian},
doi = {10.2307/2118654},
faupublication = {yes},
journal = {Annals of Mathematics},
pages = {97130},
peerreviewed = {Yes},
title = {{Wild} {Cantor} attractors exist},
volume = {143},
year = {1996}
}
@article{faucris.119932604,
abstract = {Let X ⊂ ℝ^{2} be a finite union of bounded polytopes and let T : X → X be piecewise affine and eventually expanding. Then the PerronFrobenius operator £ of T is quasicompact as an operator on the space of functions of bounded variation on ℝ^{2} and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T. In higher dimensions the result remains true under an additional generically satisfied transversality assumption.},
author = {Buzzi, Jérôme and Keller, Gerhard},
doi = {10.1017/S0143385701001341},
faupublication = {yes},
journal = {Ergodic Theory and Dynamical Systems},
note = {UnivISImport:20150309:Pub.2001.nat.dma.pma29.zetafu},
pages = {690716},
peerreviewed = {Yes},
title = {{Zeta} functions and transfer operators for multidimensional piecewise affine and expanding maps},
volume = {21},
year = {2001}
}
@article{faucris.106582564,
author = {Hofbauer, Franz and Keller, Gerhard},
doi = {10.1515/crll.1984.352.100},
faupublication = {no},
journal = {Journal für die reine und angewandte Mathematik},
pages = {100113},
peerreviewed = {unknown},
title = {{Zeta}functions and transferoperators for piecewise linear transformations},
volume = {352},
year = {1984}
}
@article{faucris.117445724,
abstract = {Given a piecewise monotone transformationT of the interval and a piecewise continuous complex weight functiong of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T, g) extends meromorphically to {∣z∣<θ^{1}} (where θ=lim ∥g^{°}T^{n1}...g^{°}Tg∥ _{∞} ^{1/n} ) and thatz is a pole of ζ if and only ifz ^{−1} is an eigenvalue of the corresponding transfer operator L. We do not assume that L leaves a reference measure invariant.},
author = {Baladi, Viviane and Keller, Gerhard},
doi = {10.1007/BF02104498},
faupublication = {yes},
journal = {Communications in Mathematical Physics},
pages = {459477},
peerreviewed = {Yes},
title = {{Zeta} functions and transfer operators for piecewise monotone transformations},
url = {http://projecteuclid.org/euclid.cmp/1104180216},
volume = {127},
year = {1990}
}