Convergence of pushforward measures for certain countably piecewise linear Markov maps

TL;DR

This paper investigates the convergence behavior of measures under certain countably piecewise linear Markov maps, introducing new symbolic dynamics techniques and proving a convergence theorem that addresses an open problem from a 2022 competition.

Contribution

The paper introduces $ ext{\ell}$-Markov partitions for countably piecewise linear maps, relates them to Markov shifts, and proves a convergence theorem for pushforward measures, addressing an open problem.

Findings

Established a convergence theorem for pushforwards of absolutely continuous measures.

Connected symbolic dynamics with countable Markov partitions.

Solved the original competition problem using the new convergence results.

Abstract

We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Mikl\'os 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-$\sigma$-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.