Unidirectionally coupled interval maps: between dynamics and statistical mechanics

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.