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

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 $\|f^*-f_{n,k}\|_1$.