Multifractal analysis for conformal graph directed Markov systems

TL;DR

This paper develops a multifractal analysis framework for conformal measures in graph directed Markov systems, applicable to various settings including continued fractions, providing insights into the measure's local scaling properties.

Contribution

It introduces a multifractal analysis method for conformal measures in GDMS, extending to complex and real continued fractions, under broad conditions.

Findings

Multifractal spectra are derived for conformal measures.

Analysis applies to a wide class of GDMS, including boundary separation cases.

Results include applications to continued fractions.

Abstract

We derive the multifractal analysis of the conformal measure (or equivalently, the invariant measure) associated to a family of weights imposed upon a (multi-dimensional) graph directed Markov system (GDMS) using balls as the filtration. This analysis is done over a large subset of the limit set. In particular, it coincides with the limit set when the GDMS under scrutiny satisfies a boundary separation condition. It also applies to more general situations such as real or complex continued fractions.