Adding machines and wild attractors - Bruin, Hendrik Pieter; Keller, Gerhard; St. Pierre, Matthias (Ergodic Theory and Dynamical Systems, 1997)
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 addingmachine-like 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.

A model for the nonautonomous Hopf bifurcation - Anagnostopoulou, Vasiliki; Jäger, Tobias; Keller, Gerhard (Nonlinearity, 2015)
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 two-step 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 measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.

An elementary proof for the dimension of the graph of the classical Weierstrass function - Keller, Gerhard (Annales de l'Institut Henri Poincaré - Probabilités Et Statistiques, 2015)
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+fracloglambdalog 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 multi-scale estimate.

An ergodic theoretic approach to mean field coupled maps - Keller, Gerhard (Fractal Geometry and Stochastics II, 2000)
We study infinite systems of globally coupled maps with permutation invariant interaction as limits of large finite-dimensional 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 SRB-measures of the finite-dimensional 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 finite-dimensional SRB-measures. We prove that this does neither happen for C3-expanding 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).
UnivIS-Import:2015-03-09:Pub.2000.nat.dma.pma29.anergo

A new estimator for information dimension with standard errors and confidence intervals - Keller, Gerhard (Stochastic Processes and their Applications, 1997)

A note on dynamical zeta functions for S-unimodal maps - Keller, Gerhard (Colloquium Mathematicum, 2000)
Let f be a nonrenormalizable S-unimodal map. We prove that f is a Collet-Eckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map.

A note on invariance principles for v. Mises' statistics - Denker, Manfred; Keller, Gerhard; Grillenberger, Christian (Metrika, 1985)
We extend Filippova's result on weak convergence of v. Mises' functionals and prove a weak invariance principle. Applications toU-statistics are given and extensions to contiguity and weakly dependent processes are briefly discussed.

A note on strange nonchaotic attractors - Keller, Gerhard (Fundamenta Mathematicae, 1996)
For a class of quasiperiodically forced time-discrete dynamical systems of two variables (θ, x) ∈ duoble-struct T sign1 × 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 Lebesgue-a.e. starting point in duoble-struct T sign1 × ℝ+. The set θ : φ(θ) ≠ 0 is meager but has full 1-dimensional Lebesgue measure. 2. The omega-limit of Lebesgue-a.e. point in duoble-struct Tsign1 × ℝ+ is Γ̄, but for a residual set of points in duoble-struct Tsign1 × 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.
UnivIS-Import:2015-03-05:Pub.1996.nat.dma.pma29.anoteo

A Sharkovskii-type theorem for minimally forced interval maps - Fabbri, Roberta; Jäger, Tobias; Johnson, Russel; Keller, Gerhard (Topological Methods in Nonlinear Analysis, 2005)
We
UnivIS-Import:2015-03-09:Pub.2005.nat.dma.pma29.ashark

A spectral gap for a one-dimensional lattice of coupled piecewise expanding interval maps - Keller, Gerhard; Liverani, Carlangelo (Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Physics, 2005)
We study one-dimensional 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.
UnivIS-Import:2015-04-20:Pub.2005.nat.dma.pma29.aspect

Automorphisms of B-free and other Toeplitz shifts - Dymek, Aurelia; Kasjan, StanisłAW; Keller, Gerhard (Ergodic Theory and Dynamical Systems, 2023)
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 B-free subshifts satisfying them, extending the work of Dymek [Automorphisms of Toeplitz B-free systems. Bull. Pol. Acad. Sci. Math. 65(2) (2017), 139-152]. Additionally, we provide an example of a B-free 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. Kułaga-Przymus and M. Lemańczyk. Springer, Cham, 2018, pp. 163-235].
CRIS-Team Scopus Importer:2023-06-30
Keywords: automorphisms group; B-free dynamics; sets of multiples; Toeplitz dynamical system; trivial centralizer

Bifurcation and Hausdorff dimension in families of chaotically driven maps with multiplicative forcing - Keller, Gerhard; Otani, Atsuya (Dynamical Systems-An International Journal, 2013)
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 one-dimensional 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 non-trivial 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.

Chaotically driven sigmoidal maps - Keller, Gerhard; Otani, Atsuya (Stochastics and Dynamics, 2017)
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
Keywords: Skew product; negative Schwarzian; attractor; Hausdorff dimension

Circular codes, loop counting, and zeta-functions - Keller, Gerhard (Journal of Combinatorial Theory Series A, 1991)
We prove a simple formula for the zeta-function 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 zeta-function of the Dyck-shift over 2N symbols, and to the zeta-function of a subshift of finite ype which is obtained from a full shift by deleting one block of arbitrary length.

Completely mixing maps without limit measure - Keller, Gerhard (Colloquium Mathematicum, 2004)
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma29.comple

Continuity properties of transport coefficients in simple maps - Howard, Phil J.; Keller, Gerhard; Klages, Rainer (Nonlinearity, 2008)
We consider families of dynamics that can be described in terms of Perron-Frobenius 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|.(log|δλ|) 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.
UnivIS-Import:2015-03-09:Pub.2008.nat.dma.zentr.contin

Coupled map lattices without cluster expansion - Keller, Gerhard; Liverani, Carlangelo (Discrete and Continuous Dynamical Systems, 2004)
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.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma29.couple
Keywords: Coupled map lattice; Exponential decay of correlations; Spatio-temporal chaos; Spectral gap; Transfer operator

Coupled map lattice via transfer operators on functions of bounded variation - Keller, Gerhard (Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), 1996)
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 non-local, but the total influence of all sites j<F NaN> 6= i on site i must be small. The main technical tool are lattice-size independent estimates of Lasota-Yorke type which show that the transfer (Perron-Frobenius) 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 time-discrete 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]...
Keywords: bounded variation map lattice via transfer operator lasota-yorke type zor zn dz main technical tool coupled system general tent map chaotic character whole system state space markov partition time-discrete dynamic weak interaction lattice-size independent estimate countable index set transfer operator approach total influence local map unpublished thesis map lattice independent chaotic action

Dynamics of B-Free sets: a view through the window - Kasjan, Stanislaw; Keller, Gerhard; Lemańczyk, Mariusz (International Mathematics Research Notices, 2019)
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.

Dynamics on the graph of the torus parametrisation - Keller, Gerhard; Richard, Christoph (Ergodic Theory and Dynamical Systems, 2016)
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

Eigenfunctions for smooth expanding circle maps - Keller, Gerhard; Rugh, Hans Henrik (Nonlinearity, 2004)
We construct a real-analytic circle map for which the corresponding Perron-Frobenius operator has a real-analytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon C1-functions.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma29.eigenf

Entropy - Warnecke, Gerald; Keller, Gerhard; Greven, Andreas (Princeton University Press, 2003)
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. In-depth 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
UnivIS-Import:2015-04-02:Pub.2003.nat.dma.pma29.entrop

Entropy of interval maps via permutations - Brandt, Christoph; Keller, Gerhard; Pompe, Bernd (Nonlinearity, 2002)
For piecewise monotone interval maps, we show that the Kolmogorov-Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.
UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pma29.entrop

Equilibrium states and Hausdorff measures for interval maps - Hofbauer, Franz; Keller, Gerhard (Mathematische Nachrichten, 1993)
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].

Equilibrium states for piecewise monotonic transformations - Hofbauer, Franz; Keller, Gerhard (Ergodic Theory and Dynamical Systems, 1982)

Equilibrium states for S-unimodal maps - Bruin, Hendrik Pieter; Keller, Gerhard (Ergodic Theory and Dynamical Systems, 1998)
For S-unimodal 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.
Keywords: equilibrium state s-unimodal map free energy uniformly hyperbolic periodic orbit log jf pressure function observable measure

Equilibrium States in Ergodic Theory - Keller, Gerhard (Cambridge University Press, 1998)
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, Sinai-Bowen-Ruelle 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.
UnivIS-Import:2015-04-02:Pub.1998.nat.dma.pma29.equili

Ergodicité et mesures invariantes pour les transformations dilatantes par morceaux d'une région bornée du plan - Keller, Gerhard (Comptes Rendus de l'Académie des Sciences - Série A: Mathématiques, 1979)

Ergodic properties of invariant measures for piecewise monotonic transformations - Hofbauer, Franz; Keller, Gerhard (Mathematische Zeitschrift, 1982)

Exponential weak Bernoulli mixing for Collet-Eckmann maps - Keller, Gerhard (Israel Journal of Mathematics, 1994)
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 lim<span class="MathJax" data-mathml="liminfn→∞<mroot>|DTn(Tc)|n</mroot>>1" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">liminfn→∞|DTn(Tc)|−−−−−−−−−√n>1, wherec is the unique critical point ofT.

Exponents, attractors and Hopf decompositions for interval maps - Keller, Gerhard (Ergodic Theory and Dynamical Systems, 1990)

Fibonacci maps re(al)visited - Keller, Gerhard; Nowicki, Tomasz (Ergodic Theory and Dynamical Systems, 1995)
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 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

Forgetful memories - van Hemmen, Jan L.; Keller, Gerhard; Kühn, Reimer (EPL - Europhysics Letters, 1988)

Generalized bounded variation and applications to piecewise monotonic transformations - Keller, Gerhard (Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1985)
We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Hölder-continuous or, more generally, of bounded p-variation.

Generalized heredity in B-free systems - Keller, Gerhard (Stochastics and Dynamics, 2021)
Let B N{1 be a primitive set, FB := ZbZ, := 1FB 0, 1Z, and denote by X the orbit closure of under the shift. We complement results on heredity of X from [Dymek et al., B-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425 5489] in two directions: In the proximal case we prove that a certain subshift X X, which coincides with X when B is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set B has light tails, as in [Dymek et al., B-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425 5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that X is always hereditary above its unique minimal (Toeplitz) subsystem . Finally, we characterize this Toeplitz subsystem as being a set X, where = 1FB for a set B that can be derived from B, and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of B-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690 2734] are heavily used.
CRIS-Team Scopus Importer:2021-04-19
Keywords: B-free dynamics; heredity; sets of multiples

How chaotic are strange non-chaotic attractors - Glendinning, Paul; Jäger, Tobias; Keller, Gerhard (Nonlinearity, 2006)
We show that the classic examples of quasiperiodically forced maps with strange non-chaotic attractors described by Grebogi et al and Herman in the mid-1980s 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 two-dimensional fibres also introduced by Grebogi et al. © 2006 IOP Publishing Ltd and London Mathematical Society.
UnivIS-Import:2015-03-09:Pub.2006.nat.dma.pma29.howcha

Interval maps with strictly contracting Perron-Frobenius operators - Keller, Gerhard (International Journal of Bifurcation and Chaos, 1999)
In this paper we give a computable criterion for a piecewise-expanding interval map T to be mixing, which at the same time not only establishes explicit bounds on the spectral gap of the associated Perron-Frobenius 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 inf|T′| > 2 and the bounds derived for them compare favorably with other estimates found in the literature.
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma29.interv

Invariant graphs for chaotically driven maps - Fadaei, Sara; Keller, Gerhard; Ghane, Fatemeh Helen (Nonlinearity, 2018)
Keywords: invariant graph; skew product; synchronization; negative Schwarzian derivative; pinch points

Ion mass and scaling effects in PIII simulation - Keller, Gerhard; Mändl, S.; Rauschenbach, B.; Rüde, Ulrich (Surface & Coatings Technology, 2001)
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 xini, depends only on the voltage and ion density and is independent of the ion mass. In this report two-dimensional particle-in-cell (PIC) simulations for PIII treatment of trench structures with different tr/√mi ratios are presented, encompassing the range of mi = 4-131 (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.
UnivIS-Import:2015-03-09:Pub.2001.tech.IMMD.lsinfs.ionmas

Irregular B -free Toeplitz sequences via Besicovitch’s construction of sets of multiples without density - Keller, Gerhard (Monatshefte für Mathematik, 2022)
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∈0,1Z is a Toeplitz sequence. Moreover our construction produces examples, for which η is not only quasi-generic 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.
CRIS-Team Scopus Importer:2022-09-16
Keywords: B-free dynamics; Density; Irregular Toeplitz sequence; Sets of multiples

Lifting measures to Markov extensions - Keller, Gerhard (Monatshefte für Mathematik, 1989)
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 anS-unimodal map with an attracting invariant Cantor set, then ∫log|T′|dμ=0 for the unique invariant measure μ on the Cantor set.   (2) IfT is piecewise invertible, iff is the Radon-Nikodym derivative ofT with respect to a σ-finite measurem, if logf has bounded distortion underT, and if μ is an ergodicT-invariant measure satisfying a certain lower estimate for its entropy, then μ≪m iffh μ (T)=Σlogf dμ.

Limit theorems for coupled interval maps - Bardet, Jean-Baptiste; Gouëzel, Sébastien; Keller, Gerhard (Stochastics and Dynamics, 2007)
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 Fourier-transfer 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.
UnivIS-Import:2015-03-09:Pub.2007.nat.dma.pma29.limitt
Keywords: Coupled map lattice; Local limit theorem; Piecewise expanding map; Spectral gap

Linear rank statistics, bounded operators, and weak convergence - Denker, Manfred; Keller, Gerhard; Puri, Madan Lal (New perspectives in theoretical and applied statistics (Bilbao, 1986), 1987)

Lyapunov exponents and complexity for interval maps - Keller, Gerhard (Lyapunov exponents (Oberwolfach, 1990), 1991)

Map lattices coupled by collisions - Keller, Gerhard; Liverani, Carlangelo (Communications in Mathematical Physics, 2009)
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 space-time decay of correlations.

Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems - Keller, Gerhard (Transactions of the American Mathematical Society, 1989)

Mathematik in den Life Sciences - Keller, Gerhard (Ulmer GmbH, 2011)
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“
Keywords: Wachstums- und Populationsmodelle • Grafische Darstellung von Daten und beschreibende Statistik; Modellierung mit Differenzialgleichungen; Wahrscheinlichkeitsrechnung; Testen und Schätzen; Korrelation und Regression; Sequence Alignment

MAXIMAL EQUICONTINUOUS GENERIC FACTORS AND WEAK MODEL SETS - Keller, Gerhard (Discrete and Continuous Dynamical Systems, 2020)
The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice L subset of G x H and compact and aperiodic window W subset of H, have the maximal equicontinuous factor (MEF) (G x H)/L, if the window is toplogically regular. This picture breaks down, when the window has empty interior, in which case the MEF is trivial, although (G x H)/L continues to be the Kronecker factor for the Mirsky measure. As this happens for many interesting examples like the square-free numbers or the visible lattice points, a weaker concept of topological factors is needed, like that of generic factors [24]. For topological dynamical systems that possess a finite invariant measure with full support (E-systems) we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. This part of the paper profits strongly from previous work by McMahon [33] and Auslander [2]. In Sections 3 and 4 we determine the MEGF of orbit closures of weak model sets and use this result for an alternative proof (of a generalization) of the fact [34] that the centralizer of any B-free dynamical system of Erdos type is trivial.
CRIS-Team WoS Importer:2020-09-11

Mixing for finite systems of coupled tent maps - Keller, Gerhard (Proceedings of the Steklov Institute of Mathematics, 1997)
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 ...
UnivIS-Import:2015-03-05:Pub.1997.nat.dma.pma29.mixing
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

Nature of weak generalized synchronization in chaotically driven maps - Keller, Gerhard; Jafri, Haider H.; Ramaswamy, R. (Physical Review E, 2013)
Weak generalized synchrony in a drive-response 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

On the asymptotic behaviour of discrete time stochastic growth processes - Keller, Gerhard; Kersting, Götz; Rösler, Uwe (Annals of Probability, 1987)
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 0-mean, square-integrable martingale difference sequence, and the states <span class="MathJax" data-mathml="Xn<0" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">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 Galton-Watson processes furnish examples for the processes under consideration.

On the asymptotic behaviour of first passage times for diffusions - Keller, Gerhard; Kersting, Götz; Rösler, Uwe (Probability Theory and Related Fields, 1988)
We investigate the asymptotic behaviour of first passage times of diffusions or birth-death 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.

On the asymptotic behaviour of solutions of stochastic differential equations - Keller, Gerhard; Kersting, Götz; Rösler, Uwe (Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1984)
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 tis a Wiener process. We clarify, under which conditions X tmay be approximated on X t→∞ by means of a deterministic function. Further the question is treated, whether X tconverges in distribution on X t→∞. We deal with the Ito-solution as well as the Stratonovitch-solution and compare both.

On the clustering conjecture for Bernoulli factors of Bernoulli shifts - Keller, Gerhard (Proceedings of the American Mathematical Society, 1991)

On the dynamics of the critical Harper map - Datta, Sandip; Jäger, Tobias; Keller, Gerhard; Ramaswamy, R. (Nonlinearity, 2004)
We examine the behaviour of typical orbits in the Harper map, a quasiperiodically driven skew-product dynamical system in two dimensions. When the map is critical, namely when all Lyapunov exponents vanish, the dynamics is parabolic.
UnivIS-Import:2015-03-09:Pub.2004.nat.dma.pma29.onthed

On the Garden of Eden theorem for B-free subshifts - Keller, Gerhard; Lemanczyk, Mariusz; Richard, Christoph; Sell, Daniel (Israel Journal of Mathematics, 2022)
We prove that on B-free 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.
CRIS-Team WoS Importer:2023-01-27

On the rate of convergence to equilibrium in one-dimensional systems - Keller, Gerhard (Communications in Mathematical Physics, 1984)
We determine the essential spectral radius of the Perron-Frobenius-operator 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 one-dimensional systems to the greatest eigenvalues of generalized Perron-Frobenius-operators of the transformations (operators which yield singular invariant measures).

On the uniqueness of equilibrium states for piecewise monotone mappings - Denker, Manfred; Keller, Gerhard; Urbański, Mariusz (Studia Mathematica, 1990)

On U-statistics and v. Mises' statistics for weakly dependent processes - Denker, Manfred; Keller, Gerhard (Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1983)

Periodic orbits for interval maps - Keller, Gerhard (Stochastic modelling in biology (Heidelberg, 1988), 1990)

Periods and factors of weak model sets - Keller, Gerhard; Richard, Christoph (Israel Journal of Mathematics, 2019)
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 G-actions 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.

Phase transitions in a piecewise expanding coupled map lattice with linear nearest neighbour coupling - Bardet, Jean-Baptiste; Keller, Gerhard (Nonlinearity, 2006)
We construct a mixing continuous piecewise linear map on [-1, 1] with the property that a two-dimensional 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 13 867-88); 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 J-R Chazottes and B Fernandez (Berlin: Springer) pp 65-94) for a detailed discussion. © 2006 IOP Publishing Ltd and London Mathematical Society.
UnivIS-Import:2015-03-09:Pub.2006.nat.dma.pma29.phaset

Piecewise monotonic transformations and exactness - Keller, Gerhard (Seminar on Probability, Rennes 1978 (French), 1978)

Practical implementation of error estimation for the correlation dimension - Frank, M.; Keller, Gerhard; Sporer, Ralph (Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 1996)
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 Grassberger-Procaccia 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 time-continuous systems, in our case the Lorenz system

Quadratic maps with maximal oscillation - Hofbauer, Franz; Keller, Gerhard (Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992), 1995)
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.

Quadratic maps without asymptotic measure - Hofbauer, Franz; Keller, Gerhard (Communications in Mathematical Physics, 1990)
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.

Random minimality and continuity of invariant graphs in random dynamical systems - Keller, Gerhard; Jäger, Tobias (Transactions of the American Mathematical Society, 2012)
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.

Random perturbations of chaotic dynamical systems: Stability of the spectrum - Blank, Michael; Keller, Gerhard (Nonlinearity, 1998)
For piecewise expanding one-dimensional 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 Perron-Frobenius 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 Sinai-Bowen-Ruelle 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 one-dimendional 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 81-107), combined with a compactness argument.
UnivIS-Import:2015-03-05:Pub.1998.nat.dma.pma29.random

Random walk in Markovian environment - Dolgopyat, Dmitry; Keller, Gerhard; Liverani, Carlangelo (Annals of Probability, 2007)
We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on ℤd. 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.
UnivIS-Import:2015-04-02:Pub.2007.nat.dma.pma29.random

Rare events, escape rates and quasistationarity: some exact formulae - Keller, Gerhard; Liverani, Carlangelo (Journal of Statistical Physics, 2009)
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.

Rare events, exponential hitting times and extremal indices via spectral perturbation - Keller, Gerhard (Dynamical Systems-An International Journal, 2012)
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.

Remarks on the linear regression approach to dimension estimation - Keller, Gerhard; Sporer, Ralph (Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), 1996)
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 real-valued 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+`Γ1 ) 2 R ...

Rigorous statistical procedures for data from dynamical systems - Denker, Manfred; Keller, Gerhard (Journal of Statistical Physics, 1986)
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.
Keywords: Dynamical systems; invariant measure; Hausdorff dimension; symmetric densities; U statistics; rank statistics

Ruelle-Perron-Frobenius spectrum for Anosov maps - Blank, Michael; Keller, Gerhard; Liverani, Carlangelo (Nonlinearity, 2002)
We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius 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 quasi-compact. (Information on the existence of a Sinai-Ruelle-Bowen 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 Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.
UnivIS-Import:2015-03-09:Pub.2002.nat.dma.pma29.ruelle

Some phase transitions in coupled map lattices - Keller, Gerhard; Künzle, Martin; Nowicki, Tomasz (Physica D-Nonlinear Phenomena, 1992)
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.

Some remarks on recent results about S-unimodal maps - Hofbauer, Franz; Keller, Gerhard (Annales de l'Institut Henri Poincaré - Physique Théorique, 1990)

Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps - Keller, Gerhard; Nowicki, Tomasz (Communications in Mathematical Physics, 1992)
We study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DT n (Tc)|≧Kλ c n for some constantsK>0 and λc>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 form<span class="MathJax" data-mathml="Sn=<msubsup>Σi=0n−1</msubsup>f(Tix)" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">Sn=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.

Spectrum of weak model sets with Borel windows - Keller, Gerhard; Richard, Christoph; Strungaru, Nicolae (Canadian Mathematical Bulletin-Bulletin Canadien De Mathematiques, 2022)
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 measure-theoretic 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.
CRIS-Team WoS Importer:2022-08-26

Stability index for chaotically driven concave maps - Keller, Gerhard (Journal of the London Mathematical Society-Second Series, 2014)

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors - Keller, Gerhard (AIMS Journal, 2017)
Skew product systems with monotone one-dimensional 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

Stability of the spectrum for transfer operators - Keller, Gerhard; Liverani, Carlangelo (Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 1999)
UnivIS-Import:2015-03-05:Pub.1999.nat.dma.pma29.stabil

Stationary measures for randomly chosen maps - Burton, Robert M.; Keller, Gerhard (Journal of Theoretical Probability, 1993)
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 self-reinforcing random walks.
Keywords: Random transformation; iterated function system; weak Bernoulli; self-reinforcing random walk

Stochastically stable globally coupled maps with bistable thermodynamic limit - Bardet, Jean-Baptiste; Keller, Gerhard; Zweimüller, Roland (Communications in Mathematical Physics, 2009)
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, finite-size coupled systems always have a unique invariant probability density which is strictly positive and analytic, and all finite-size systems exhibit exponential decay of correlations. 2) For the same range of parameters, the self-consistent Perron-Frobenius operator which captures essential aspects of the corresponding infinite-size 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.

Stochastic perturbations of some strange attractors - Keller, Gerhard (Dynamical System and Chaos, Proceedings of the Sitges Conference on Statistical Mechanics Sitges, 1982)

Stochastic stability in some chaotic dynamical systems - Keller, Gerhard (Monatshefte für Mathematik, 1982)
For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator 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 Perron-Frobenius-operators.

Stochastic stability of some one-dimensional dynamical systems - Keller, Gerhard (Ergodic Theory and Related Topics. Proceedings of the Conference held in Vitte/Hiddensee, 1982)

Stochastic stability versus localization in one-dimensional chaotic dynamical systems - Blank, Michael; Keller, Gerhard (Nonlinearity, 1997)
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 so-called 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.

Synchronization versus stability of the invariant distribution for a class of globally coupled maps - Bálint, Péter; Keller, Gerhard; Mincsovicsne Selley, Fanni; Tóth, Imre Péter (Nonlinearity, 2018)
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

Tautness of sets of multiples and applications to B-free dynamics - Keller, Gerhard (Studia Mathematica, 2019)
For any set ⊆ℕ=1,2,… one can define its emphset 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 shift-orbit closure Xη⊆0,1ℤ 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 one-point 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 0,1ℤ, then also the coordinate-wise 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.

The central limit theorem for the normalized sums of the MAI for SSMA communication systems using spreading sequences of Markov chains - Fujisaki, Hiroshi; Keller, Gerhard (IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, 2006)
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 0-th 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.
Keywords: sliding block code; Markov chain; a(N)-mixing; central limit theorem (CLT); bit error probability; spread spectrum multiple access (SSMA) communication system

The Denjoy type of argument for quasiperiodically forced circle diffeomorphisms - Keller, Gerhard; Jäger, Tobias (Ergodic Theory and Dynamical Systems, 2006)

Topological and measurable dynamics of Lorenz maps - Keller, Gerhard; St. Pierre, Matthias (Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, 2001)
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.
UnivIS-Import:2015-04-20:Pub.2001.nat.dma.pma29.topolo

Transfer operators for coupled map lattices - Keller, Gerhard; Künzle, Martin (Ergodic Theory and Dynamical Systems, 1992)
Let L denote a finite or infinite one-dimensional 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) Sε: 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 single-component-transformations 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 T-invariant measures with absolutely continuous finite-dimensional marginals.

Unidirectionally coupled interval maps: between dynamics and statistical mechanics - Keller, Gerhard; Zweimüller, Roland (Nonlinearity, 2002)
We study infinite systems of unidirectionally coupled interval maps on the lattice d. It is assumed that the single-site map τ is piecewise C2 expanding, mixing and satisfies stably a Lasota-Yorke-type 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 time-invariant probability measures whose finite-dimensional conditional distributions are absolutely continuous w.r.t. the corresponding finite-dimensional 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.

Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension - Keller, Gerhard; Liverani, Carlangelo (Communications in Mathematical Physics, 2006)
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.

Un théorème de la limite centrale pour une classe de transformations monotones par morceaux - Keller, Gerhard (Comptes Rendus de l'Académie des Sciences - Série A: Mathématiques, 1980)

Wild Cantor attractors exist - Bruin, Hendrik Pieter; Keller, Gerhard; Nowicki, Tomasz; van Strien, Sebastian (Annals of Mathematics, 1996)

Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps - Buzzi, Jérôme; Keller, Gerhard (Ergodic Theory and Dynamical Systems, 2001)
Let X ⊂ ℝ2 be a finite union of bounded polytopes and let T : X → X be piecewise affine and eventually expanding. Then the Perron-Frobenius operator £ of T is quasicompact as an operator on the space of functions of bounded variation on ℝ2 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.
UnivIS-Import:2015-03-09:Pub.2001.nat.dma.pma29.zetafu

Zeta-functions and transfer-operators for piecewise linear transformations - Hofbauer, Franz; Keller, Gerhard (Journal für die reine und angewandte Mathematik, 1984)

Zeta functions and transfer operators for piecewise monotone transformations - Baladi, Viviane; Keller, Gerhard (Communications in Mathematical Physics, 1990)
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°Tn-1...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.