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Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

Nathan Dalaklis

TL;DR

The paper develops a comprehensive multifractal formalism for level sets of Birkhoff averages on cofinitely regular, finitely irreducible conformal graph directed Markov systems under the strong open set condition. By introducing a two-parameter pressure $P(t,q)$ and the Manhattan region $D$, it proves the existence and analyticity of a real-analytic spectrum $(t(\xi),q(\xi))$ solving $P(t,q)=q\partial_qP(t,q)=\xi$, with $t(\xi)$ equaling the Hausdorff dimension of the level sets $J(\xi)$ and their projections $J_1(\xi)$. The authors establish tightness of Gibbs states, zero-temperature limits yielding a universal minimal exponent $\xi_{\min}$, and a volume-lemma-based derivation of the multifractal spectrum; they also show the spectrum is neither strictly convex nor concave for infinite alphabets and illustrate the theory with Lyapunov, Gauss map, and Lüroth expansion examples. Overall, the work provides a rigorous, analytic bridge between thermodynamic formalism and geometric multifractal spectra in a broad class of dynamical systems, with concrete applications to number-theoretic expansions.

Abstract

Let $Φ= \{φ_e\}_{e\in E}$ be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation $E_A^{\infty}$ and limit set $J$. Under a mild condition on the system, we give a multifractal analysis of level sets of Birkhoff averages with respect to Hausdorff dimension for a large family of functions. We then apply these results to a few examples in the case of both $E$ finite and $E$ countably infinite.

Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

TL;DR

The paper develops a comprehensive multifractal formalism for level sets of Birkhoff averages on cofinitely regular, finitely irreducible conformal graph directed Markov systems under the strong open set condition. By introducing a two-parameter pressure and the Manhattan region , it proves the existence and analyticity of a real-analytic spectrum solving , with equaling the Hausdorff dimension of the level sets and their projections . The authors establish tightness of Gibbs states, zero-temperature limits yielding a universal minimal exponent , and a volume-lemma-based derivation of the multifractal spectrum; they also show the spectrum is neither strictly convex nor concave for infinite alphabets and illustrate the theory with Lyapunov, Gauss map, and Lüroth expansion examples. Overall, the work provides a rigorous, analytic bridge between thermodynamic formalism and geometric multifractal spectra in a broad class of dynamical systems, with concrete applications to number-theoretic expansions.

Abstract

Let be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation and limit set . Under a mild condition on the system, we give a multifractal analysis of level sets of Birkhoff averages with respect to Hausdorff dimension for a large family of functions. We then apply these results to a few examples in the case of both finite and countably infinite.
Paper Structure (17 sections, 38 theorems, 196 equations, 4 figures)

This paper contains 17 sections, 38 theorems, 196 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries for CGDMSs
  4. Topological Pressure
  5. The Manhattan Region
  6. Derivatives of Pressure
  7. Conformality of Measures
  8. Tightness and Zero-Temperature Style Limits
  9. Tightness and Prohorov's Theorem
  10. Minimizing F-exponents
  11. Multifractal Analysis of F-exponents
  12. Derivation of the Multifractal Spectrum
  13. Shape of F-Exponent Spectra
  14. Applications
  15. Lyapunov Spectrum
  16. ...and 2 more sections

Key Result

Theorem 1.1.1

Let $\Phi = \{\phi_e\}_{e\in\mathbb{N}}$ be a cofinitely regular finitely irreducible CGDMS satisfying the SOSC. Let $F$ be a strictly positive Hölder family of potentials that are either comparable to $\log\Phi'$ (see Lemma lem:Dopen) or bounded. Let $D$ be the Manhattan region of the associated pr where $\tilde{\mu}_{t,q}$ is the unique shift invariant Borel probability measure equivalent to the

Figures (4)

  • Figure 1: Graph of $\mathop{\mathrm{HD}}\limits(J(\xi))$ for Example \ref{['ex:Lu']}.
  • Figure 2: Graph of Numerical Solution of $t(\xi)$ for Example \ref{['ex:LG3_Parity']} via MATLAB
  • Figure 3: Graph of Numerical Solution of $t(\xi)$ for Example \ref{['ex:LG3_23']} via MATLAB
  • Figure 4: Graph of $t(\xi)$ for Example \ref{['ex:Lu']} for $\xi\in(\log 2,10))$

Theorems & Definitions (87)

  • Theorem 1.1.1
  • Proposition 1.1.2: c.f. jenkinsonZeroTemperatureLimits2005 Lemma 2
  • Proposition 1.1.3
  • Definition 1.2.1
  • Definition 1.2.2
  • Definition 1.2.3
  • Definition 1.2.4
  • Definition 1.2.5
  • Definition 1.2.6
  • Definition 1.2.7
  • ...and 77 more