2025/08/25 by Kalocsai, Zoltán
#37A25 #37A30 #37A40 #37B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 37A50 Secondary: 37A05
paper · doi:10.48550/arxiv.2508.18172
We study piecewise linear Markov maps, with countable Markov partitions, inspired by a problem of the Miklós Schweitzer competition in 2022. We introduce ℓ-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.