Transition to anomalous dynamics in a simple random map

TL;DR

This paper analyzes a simple random dynamical system combining expanding and contracting maps, revealing a transition to anomalous dynamics with infinite invariant density and weak ergodicity breaking at a critical probability.

Contribution

It provides an analytical and numerical study of the invariant density and autocorrelation in a random map, identifying a transition to anomalous dynamics at a critical probability.

Findings

Transition from chaotic to regular dynamics at critical probability p_c

Presence of infinite invariant density at p_c

Power law decay of correlations in the anomalous regime

Abstract

The famous Bernoulli shift (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability $p$, and the contracting one with probability $1-p$, gives a prototype of a random dynamical system. Here we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of $p$. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability $p_c$, defined by a zero Lyapunov exponent. This anomalous dynamics is characterised by an infinite invariant density, weak ergodicity breaking and power law correlation decay.