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 $f^{\ast }$, 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 $f_{n,k}\to f^{\ast }$ 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 $\Vert f^{\ast }-f_{n,k}{\Vert }_1$.

19 pages, 9 figures