Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding R\'{e}nyi interval maps with countably many branches

TL;DR

This paper develops a thermodynamic formalism and analyzes the multifractal spectrum of Birkhoff averages for complex Rényi interval maps, providing new formulas and confirming existing conjectures in the field.

Contribution

It introduces a strengthened variational formula and a detailed multifractal analysis for non-uniformly expanding Rényi maps with countably many branches, advancing the theoretical understanding.

Findings

Strengthened conditional variational formulas.

Detailed analysis of Hausdorff dimension related to Khinchin exponents.

Confirmed the conjecture of Jaerisch and Takahasi.

Abstract

In this paper, we study the multifractal spectrum of Birkhoff averages for non-uniformly expanding R\'{e}nyi interval maps with countably many branches. Our main theorem substantially strengthens conditional variational formulas established by Jaerisch and Takahasi. Furthermore, our results enable a detailed analysis of Khinchin exponents and arithmetic means of backward continued fraction expansions in terms of the Hausdorff dimension. We also give a positive answer to the conjecture of Jaerisch and Takahasi. In addition, we develop the thermodynamic formalism for non-uniformly expanding R\'{e}nyi interval maps with countably many branches.