Probabilistic models of multidimensional piecewise expanding mappings
Because strongly chaotic systems exhibit diffusion-like behavior in the sense of central limit theorems, the author of this paper works toward the goal of finding more precise statements about the similarity to probabilistic diffusion by studying how the evolution of probability densities under \(d\)-dimensional piecewise expanding mappings can be modeled by Markov processes with smooth transition probabilities. The author investigates a class of discrete time dynamical systems given by: \[ X_{n+ 1}= X_n+ f(X_n, y_n),\quad y_{n+ 1}= y_n+ g(X_n, y_n), \] where \(X_n\in \mathbb{R}^d\) and \(y_n\) lies on an \(\ell\)-dimensional torus, and where \(f\) and \(g\) are integer-periodic with respect to \(X_n\). One then regards \(X_n\) as an observable, \(y_n\) as a hidden variable, and looks for a probabilistic system which approximates the behavior of \(X_n\) as \(n\to \infty\). To do this, the author fixes probability densities for \(X_0\) and \(y_0\) and treats \(X_n\) as a random variable with density function \(\rho^d_n\). Then he searches for a stochastic process whose long term density \(\rho_n\) matches the long term behavior of \(X_n\) in both microscopic \((\sim 1)\) and macroscopic \((\sim\sqrt n)\) scales. The author establishes that, in the general case, the decay \(|\rho^d_n- \rho_n|\) in the \(L^1(\mathbb{R}^d)\) norm is faster than \(n^{-\gamma}\) for all \(\gamma< 1/2\). This probabilistic approximation is of global and local type.
multidimensional piecewise expanding maps
scientific article; zbMATH DE number 796342
scientific article