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arXiv · 2508.18172

Convergence of pushforward measures for certain countably piecewise linear Markov maps

Zoltán Kalocsai
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Abstract

We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Miklós Schweitzer competition in 2022. We introduce $\ell$-Markov partitions and apply ideas of symbolic dynamics to our systems, relating them to Markov shifts. We survey how the Frobenius--Perron operators of these systems can be represented by matrices, and adapt results to countable alphabets. We apply these statements to prove a convergence theorem on the pushforwards of absolutely continuous measures. This enables us to prove a variety of useful ergodic properties of our maps and study even non-$σ$-finite absolutely continuous invariant measures. We explain how our results are not implied by previous ones and apply the convergence theorem to solve the original problem in the competition.

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Zoltán Kalocsai. 2025-08-25. Convergence of pushforward measures for certain countably piecewise linear Markov maps. https://arxiv.org/abs/2508.18172

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