Multifractal analysis for Markov interval maps with countably many branches

TL;DR

This paper extends multifractal analysis to expanding interval maps with countably many branches, exploring how different coding conditions affect the multifractal spectrum and generalizing previous finite-branch results.

Contribution

It introduces a new analysis framework for countably branched maps, broadening the scope of multifractal decompositions beyond finite and full shift cases.

Findings

Generalizes multifractal analysis to countably many branches

Identifies different behaviors based on coding entropy and derivative bounds

Shows that finite entropy coding leads to distinct multifractal properties

Abstract

We study multifractal decompositions based on Birkhoff averages for sequences of functions belonging to certain classes of symbolically continuous functions. We do this for an expanding interval map with countably many branches, which we assume can be coded by a topologically mixing countable Markov shift. This generalises previous work on expanding maps with finitely many branches, and expanding maps with countably many branches where the coding is assumed to be the full shift. When the infimum of the derivative on each branch approaches infinity in the limit, we can directly generalise the results of the full countable shift case. However, when this does not hold, we show that there can be different behaviour, in particular in cases where the coding has finite topological entropy.