Ulam's method for computing stationary densities of invariant measures for piecewise convex maps with countably infinite number of branches

TL;DR

This paper develops an Ulam method to approximate the stationary density of invariant measures for piecewise convex maps with infinitely many branches, proving convergence of the approximations in various norms.

Contribution

It introduces a novel approach combining Ulam's method with convergence analysis for maps with countably infinite branches, extending previous finite-branch techniques.

Findings

Convergence of the approximated densities to the true density in L^1 and almost everywhere.

Construction of finite-branch maps approximating infinite-branch maps.

Numerical examples demonstrating error decay with increasing parameters.

Abstract

Let $\tau: I=[0, 1]\to [0, 1]$ be a piecewise convex map with countably infinite number of branches. In \cite{GIR}, the existence of absolutely continuous invariant measure (ACIM) $\mu$ for $\tau$ and the exactness of the system $(\tau, \mu)$ has been proven. In this paper, we develop an Ulam method for approximation of $f^*$, the density of ACIM $\mu$. We construct a sequence $\{\tau_n\}_{n=1}^\infty$ of maps $\tau_n: I\to I$ s. t. $\tau_n$ has a finite number of branches and the sequence $\tau_n$ converges to $\tau$ almost uniformly. Using supremum norms and Lasota-Yorke type inequalities, we prove the existence of ACIMs $\mu_n$ for $\tau_n$ with the densities $f_n$. For a fixed $n$, we apply Ulam's method with $k$ subintervals to $\tau_n$ and compute approximations $f_{n,k}$ of $f_n$. We prove that $f_{n,k}\to f^*$ as $n\to \infty, k\to \infty,$ both a.e. and in $L^1$. We provide examples of piecewise convex maps $\tau$ with countably infinite number of branches, their approximations $\tau_n$ with finite number of branches and for increasing values of parameter $k$ show the errors $\|f^*-f_{n,k}\|_1$.