Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches

Yuya Arima

TL;DR

The paper advances the thermodynamic formalism for non-uniformly expanding Rényi interval maps with countably many branches and uses it to derive a sharp, comprehensive description of the Birkhoff spectrum for a broad class of potentials. By introducing the two-parameter family of potentials $-q\phi-b\log|f'|$ on induced and original systems, it proves the real-analyticity of the pressure, the existence and uniqueness of equilibrium measures, and a conditional variational principle that yields exact formulas for $b(\alpha)$ across the spectrum. A central result is that $b(\alpha)=\delta$ for all $\alpha$ in the interval $A$, while outside $A$ the spectrum is given by entropy/Lyapunov-exponent ratios of carefully chosen measures, with detailed monotonicity and analyticity properties depending on $\mathfrak{R}$ and regularity assumptions. The framework is applied to backward continued fraction expansions, solving conjectures and clarifying Khinchin exponents in terms of Hausdorff dimension, thereby connecting multifractal analysis with Diophantine-type questions and expanding the toolkit for non-uniformly hyperbolic systems with countable branches.

Abstract

In this paper, we study the multifractal spectrum of Birkhoff averages for non-uniformly expanding Rényi 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 Rényi interval maps with countably many branches.

Thermodynamic formalism and multifractal analysis of Birkhoff averages for non-uniformly expanding Rényi interval maps with countably many branches

TL;DR

The paper advances the thermodynamic formalism for non-uniformly expanding Rényi interval maps with countably many branches and uses it to derive a sharp, comprehensive description of the Birkhoff spectrum for a broad class of potentials. By introducing the two-parameter family of potentials on induced and original systems, it proves the real-analyticity of the pressure, the existence and uniqueness of equilibrium measures, and a conditional variational principle that yields exact formulas for across the spectrum. A central result is that for all in the interval , while outside the spectrum is given by entropy/Lyapunov-exponent ratios of carefully chosen measures, with detailed monotonicity and analyticity properties depending on and regularity assumptions. The framework is applied to backward continued fraction expansions, solving conjectures and clarifying Khinchin exponents in terms of Hausdorff dimension, thereby connecting multifractal analysis with Diophantine-type questions and expanding the toolkit for non-uniformly hyperbolic systems with countable branches.

Abstract

In this paper, we study the multifractal spectrum of Birkhoff averages for non-uniformly expanding Rényi 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 Rényi interval maps with countably many branches.

Paper Structure

This paper contains 16 sections, 51 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Application of the main theorem to backward continued fraction expansions
  3. Outline of the paper
  4. Preliminary
  5. thermodynamic formalism
  6. Convergence of equilibrium measures
  7. Multifractal analysis
  8. Conditional variational principle
  9. The flat part and lower bound of $b(\alpha)$
  10. Properties of the function $(b,q)\mapsto p(b,q)+q\alpha$
  11. Monotonicity of the Birkhoff spectrum
  12. Analysis of the set $\mathcal{G}$ under the condition $\mathfrak{R}=0$
  13. Analysis of the set $\mathcal{G}$ under the condition $\mathfrak{R}=\infty$
  14. Analysis of the set $\mathcal{G}$ under the condition $0<\mathfrak{R}<\infty$
  15. Proof of Theorem \ref{['thm main']}.
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $f$ be a non-uniformly expanding Rényi interval map with countably many branches having the admissible induced map $\tilde{f}$ and let $\phi\in\mathcal{R}$ satisfy (R). We also assume that $f$ satisfies (G). Then, for all $\alpha\in A$ we have $b(\alpha)=\delta$. Furthermore, we have the followi

Theorems & Definitions (86)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • ...and 76 more