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

Rafael Lucena

doi:10.3934/dcds.2025108

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2026, Volume 46: 475-504. Doi: 10.3934/dcds.2025108

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

Rafael Lucena^,

Federal University of Alagoas, Brazil

^*Corresponding author: Rafael Lucena
^*Corresponding author: Rafael Lucena

In loving memory of my dear friend Fellyppe Carlos Santos de Lima.

Received: February 2025

Revised: June 2025

Early access: July 18, 2025

Published online: July 18, 2025

The author is supported by FAPEAL (Alagoas-Brazil) Grant E:60030.0000002330/2022 and CNPq (Brazil) Grant Alagoas Dinâmica: 409198/2021-8.

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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.

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Mathematics Subject Classification: Primary: 37A25, 37A30; Secondary: 37H05.

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References

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