Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

Abstract

In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $\tau$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $\tau$, ${\tau }^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $\mu$ for $\tau$, the exactness of the dynamical system $(I,\tau ,\mu )$ and the quasi-compactness of Frobenius-Perron operator $P_{\tau }$ induced by $\tau$. The last fact implies a multitude of strong ergodic properties of $\tau$.

23 pages, 6 figures