Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces

Yuya Arima

TL;DR

The paper develops a higher-dimensional multifractal analysis for cusp windings of geodesic flow on hyperbolic surfaces with $m\ge1$ cusps. It defines the $m$-dimensional cusp-winding spectrum $b(\boldsymbol{\alpha})=\dim_H J(\boldsymbol{\alpha})$ and proves a conditional variational principle, with $b$ realized as a supremum of entropy-to-Lyapunov-ratio over invariant measures, extending prior $m=1$ results. Using a countable Markov shift with a finitely primitive matrix and thermodynamic formalism, it proves that $b$ is real-analytic on $(0,\infty)^m$ and establishes a Bowen-type relation linking the spectrum to topological pressure via a zero of $p(\boldsymbol{\alpha},\boldsymbol{q},b(\boldsymbol{\alpha}))$. In the $m=1$ case, it shows strict monotonicity of the spectrum and a maximal limit, and it demonstrates that the irregular set shares the same Hausdorff dimension as the conical limit set. Together, these results provide a comprehensive, higher-dimensional multifractal picture for cusp windings on hyperbolic surfaces and open avenues for further thermodynamic analyses in similar symbolic dynamics settings.

Abstract

We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with $m \geq 1$ cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on $\mathbb{R}_{>0}^m$ is real analytic. To prove the main theorem we use a countable Markov shift with a finitely primitive transition matrix and thermodynamic formalism.

Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces

TL;DR

The paper develops a higher-dimensional multifractal analysis for cusp windings of geodesic flow on hyperbolic surfaces with cusps. It defines the -dimensional cusp-winding spectrum and proves a conditional variational principle, with realized as a supremum of entropy-to-Lyapunov-ratio over invariant measures, extending prior results. Using a countable Markov shift with a finitely primitive matrix and thermodynamic formalism, it proves that is real-analytic on and establishes a Bowen-type relation linking the spectrum to topological pressure via a zero of . In the case, it shows strict monotonicity of the spectrum and a maximal limit, and it demonstrates that the irregular set shares the same Hausdorff dimension as the conical limit set. Together, these results provide a comprehensive, higher-dimensional multifractal picture for cusp windings on hyperbolic surfaces and open avenues for further thermodynamic analyses in similar symbolic dynamics settings.

Abstract

We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on is real analytic. To prove the main theorem we use a countable Markov shift with a finitely primitive transition matrix and thermodynamic formalism.
Paper Structure (9 sections, 26 theorems, 90 equations)

This paper contains 9 sections, 26 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaies
  3. The Bowen-Series map and multi-cusp winding process
  4. Thermodynamic formalism
  5. Improved conditional variational principle for the dimension spectrum
  6. Convergence of the topological pressure and Bowen's formula
  7. Relationship between the topological pressure and dimension spectrum
  8. Regularity of the dimension spectrum
  9. Proof of Proposition \ref{['Proposition m=1 case']}

Key Result

Theorem 1.1

Let $G$ be a generalized Schottky group with $m\geq1$ parabolic generators. Then the following holds: (1) For $\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_m)\in(0,\infty)^m$ we have (2) There exists $\mu\in M(\tilde{f})$ such that (3) The dimension spectrum $b$ is real-analytic on $(0,\infty)^m$. (4) We have $\dim_H(J_{\text{ir}})=\dim_H(\Lambda(G))$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: jaerisch2021mixed
  • Definition 2.6
  • Definition 2.7: mauldin2003graph
  • ...and 41 more