Abstract The Pelikan random trajectories x t ∈ [ 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 = 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 ∈ [ 2 − n − 1 , 2 − n [ partitioning the interval x ∈ [ 0 , 1 [ : the dynamics for the weights π t ( n ) of these intervals corresponds to a biased random walk on the half-infinite lattice n ∈ { 0 , 1 , 2 , . . + ∞ } 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 = ∑ l = 1 + ∞ σ l ( t ) 2 l , where the dynamics for the half-infinite lattice l = 1 , 2 , . . of the binary variables σ l ( t ) ∈ { 0 , 1 } can be rephrased in terms of two global variables: zt corresponds to a biased random walk on the half-infinite lattice z ∈ { 0 , 1 , 2 , … + ∞ } that may remain at the origin z = 0 with probability p, while F t ∈ { 0 , 1 , 2 , … , t } counts the number of time-steps τ ∈ [ 0 , t − 1 ] where z τ + 1 = 0 = z τ 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 1 2 < p < 1 as well as at the intermittent critical point p c = 1 2 towards the non-chaotic region 0 < p < 1 2 .
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Jan 1, 2025
Essam B Moustafa + 5
Open Access