Transition to anomalous dynamics in a simple random map

The famous doubling map (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 pc, defined by a zero Lyapunov exponent. This anomalous dynamics is characterized by an infinite invariant density, weak ergodicity breaking, and power-law correlation decay.