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.