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

This is the authors' abstract. We don't add key points for this paper.

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