Coupled Map Lattices (CML) are a kind of dynamical systems that appear naturally in some contexts, like the discretization of partial differential equations, and as a simple model of coupling between nonlinear systems. The coupling creates new and rich properties, that has been the object of intense investigation during the last decades. In this work we have two goals: first we give a nontechnical introduction to the theory of invariant measures and equilibrium in dynamics (with analogies with equilibrium in statistical mechanics) because we believe that sometimes a lot of interesting problems on the interface between physics and mathematics are not being developed simply due to the lack of a common language. Our second goal is to make a small contribution to the theory of equilibrium states for CML. More specifically, we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. The goal is to show that we start in a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the dynamics of most orbits coupling. Associating a symbolic dynamics with symbols +1, 0 and -1 to the system we describe a transition characterized by an asymptotic state composed only of symbols +1 or only of symbols -1. We rigorously prove our assertions and provide numerical experiments with two goals: first, as illustration of our rigorous results and second, to motivate some conjectures concerning the problem and some of its possible variations.
Extensive bounds on the topological entropy of repellers in piecewise expanding coupled map lattices
