Fair Measures for Countable-to-one Maps

TL;DR

This paper extends the concept of fair measures to transitive countable state Markov shifts, certain interval maps, and graph maps, providing a unified framework for understanding preimage distributions in these dynamical systems.

Contribution

It generalizes the notion of fair measures to broader classes of dynamical systems, including countable Markov shifts and graph maps, beyond previous limited contexts.

Findings

Extended fair measure concept to countable Markov shifts

Applied results to Markov and mixing interval maps

Explored fair measures for graph maps

Abstract

In this paper we generalize the recently introduced concept of fair measure (M. Misiurewicz and A. Rodrigues, Counting preimages. Ergod. Th. & Dynam. Sys. 38 (2018), no. 5, 1837 -- 1856). We study transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Finally, we move beyond the interval and look for fair measures for graph maps.