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

2024/06/28 by Gora, Pawel, Rajput, Aparna · 1 citation
#37A05 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.19929

Abstract

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 \citehofbauer1982 and \citekeller1985, 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.

Cited by

• Statistical stability for systems semi-conjugate to pre-piecewise convex or expanding maps with countably many branches

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