Broken Ergodicity and Glassy Behavior in a Deterministic Chaotic Map

Abstract

A network of $N$ elements is studied in terms of a deterministic globally coupled map which can be chaotic. There exists a range of values for the parameters of the map where the number of different macroscopic configurations $N\left(N\right)$ is very large, $N\left(N\right)\sim \exp \sqrt{c\left(a\right)N}$, and there is violation of self-averaging. The time averages of functions, which depend on a single element, computed over a time $T$, have probability distributions that for any $N$ do not collapse to a delta function, for increasing $T$. This happens for both chaotic and regular motion, i.e., positive or negative Lyapunov exponent.