2025/12/19 by Vanessa Barros, Barros, Vanessa, Adriana Coutinho +1
Economics, Econometrics and Finance · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2512.18050
openalex publication_date 2025/12/19 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28
In this paper, we investigate the asymptotic behavior of the shortest distance between observed orbits in two distinct dynamical systems. Given two measure-preserving transformations (X, T, μ) and (X, S, η) and a Lipschitz observation function f, we define \widehatmnf(x,y) = mini=0,…,n-1 d(f(Ti x), f(Si y)). %Under suitable mixing assumptions, we show that the asymptotic rate of decay of \widehatmnf(x,y) is governed by the correlation dimensions of the pushforward measures f_*μ and f_*η. Under suitable mixing assumptions, we show that the asymptotic rate of decay of \widehatmnf(x,y) is governed by the symmetric Rényi divergence of the pushforward measures f_*μ and f_*η. Our results generalize previous work that consider either a single system or the unobserved case. In addition, we discuss the extension of these results to random dynamical systems and illustrate the applicability of the approach with an example.