  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.

Chaotic Dynamics;Statistical Mechanics;Dynamical Systems