Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region

Abstract

The Pelikan random trajectories $x_t \in [0,1[$ are generated by choosing the chaotic doubling map $x_{t+1}=2 x_t [mod 1]$ with probability $p$ and the non-chaotic half-contracting map $x_{t+1}=\frac{x_t}{2}$ with probability $(1-p)$. We compute various dynamical observables as a function of the parameter $p$ via two perspectives. In the first perspective, we focus on the closed dynamics within the subspace of probability densities that remain constant on the binary-intervals $x \in [ 2^{-n-1}, 2^{-n}[$ partitioning the interval $x \in [0,1[$ : the dynamics for the weights $\pi_t(n)$ of these intervals corresponds to a biased random walk on the half-infinite lattice $n \in \{0,1,2,..+\infty\}$ with resetting occurring with probability $p$ from the origin $n=0$ towards any site $n$ drawn with the distribution $2^{-n-1}$. In the second perspective, we study the Pelikan dynamics for any initial condition $x_0$ via the binary decomposition $x_t = \sum_{l=1}^{+\infty} \frac{\sigma_l (t)}{2^l} $, where the dynamics for the half-infinite lattice $l=1,2,..$ of the binary variables $\sigma_l(t) \in \{0,1\}$ can be reformulated in terms of two global variables : $z_t$ corresponds to a biased random walk on the half-infinite lattice $z \in \{0,1,2,..+\infty\}$ that may remain at the origin $z=0$ with probability $p$, while $F_t \in \{0,1,2,..t\}$ counts the number of time-steps $\tau \in [0,t-1]$ where $z_{\tau+1}=0=z_{\tau}$ and represents the number of the binary coefficients of the initial condition that have been erased. We discuss typical and large deviations properties in the chaotic region $\frac{1}{2}<p<1 $ as well as at the intermittent critical point $p_c=\frac{1}{2}$ towards the non-chaotic region $0<p<\frac{1}{2}$.