Quasi-compactness and statistical properties for discontinuous systems semi-conjugate to piecewise convex maps with countably many branches

TL;DR

This paper proves the quasi-compactness of transfer operators for certain discontinuous skew product systems semi-conjugate to infinite-branch convex maps, and introduces a new invariant measure with bounded variation disintegration.

Contribution

It establishes quasi-compactness for transfer operators of these complex systems and introduces a novel invariant measure with bounded variation disintegration.

Findings

Transfer operators are quasi-compact for the systems studied.

Existence of an invariant measure with disintegration of bounded variation.

Systems admit discontinuities confined to countable fibers.

Abstract

In this paper, we establish the quasi-compactness of the transfer operator associated with skew product systems that are semi-conjugate to piecewise convex maps with a countably infinite number of branches. These non-invertible skew products admit discontinuities, with the critical set confined to a countable collection of fibers. Furthermore, we demonstrate that such systems possess an invariant measure whose disintegration along the fibers exhibits bounded variation; a concept introduced and developed in this work.