An iterated random function with Lipschitz number one

2025/06/27 by Abrams, Aaron, Landau, Henry, Landau, Zeph +2
#37A10 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.22420

Abstract

Consider the set of functions f_θ(x)=|θ-x| on ℝ. Define a Markov process that starts with a point x₀ ∈ ℝ and continues with x_k+1=f_θ_k+1(x_k) with each θ_k+1 picked from a fixed bounded distribution μ on ℝ⁺. We prove the conjecture of G. Letac that if μ is not supported on a lattice, then this process has a unique stationary distribution π_μ and any distribution converges under iteration to π_μ (in the weak-^* topology). We also give a bound on the rate of convergence in the special case that μ is supported on a two-point set. We hope that the techniques will be useful for the study of other Markov processes where the transition functions have Lipschitz number one.

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