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Multifractal analysis of the convergence exponents for the digits in $d$-decaying Gauss like dynamical systems

Kunkun Song, Mengjie Zhang

TL;DR

The paper addresses the multifractal analysis of convergence exponents for digits in $d$-decaying Gauss-like iIFS on $[0,1]$, introducing the convergence exponents $\tau_1(x)$ and $\tau_2(x)$ for digits and weighted products. It leverages conformal dynamics, pressure theory, and Cantor-type constructions to determine the Hausdorff dimension spectra of the corresponding level sets, showing that $\\dim_H E(\alpha)=\\dim_H E(\alpha,\{t_i\})=1/d$ for $0\le\alpha<\infty$, and revealing a piecewise spectrum for nondecreasing digits restricted to $\\Lambda$ with a threshold at $\\Sigma_t=\sum t_i$. Specifically, for $\\Lambda$, the spectrum is $0$ when $0\le\alpha<\\Sigma_t$, $(\\alpha-\\Sigma_t)/(d\\alpha)$ for $\\Sigma_t\le\alpha<\infty$, and $1/d$ at $\\alpha=\\infty$, with $F$ and $G$ sharing the same dimensional behavior. The work also connects these spectra to Lebesgue conformality and invariant measures, enriching the multifractal theory for infinite conformal systems and providing tools for analyzing digit-growth and restricted-growth sets in such systems.

Abstract

Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence of the digits $\{a_n(x)\}_{n\geq1}$ and the weighted products of distinct digits with finite numbers respectively, and then calculate the Hausdorff dimensions of the intersection of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are non-decreasing in such iterated function systems.

Multifractal analysis of the convergence exponents for the digits in $d$-decaying Gauss like dynamical systems

TL;DR

The paper addresses the multifractal analysis of convergence exponents for digits in -decaying Gauss-like iIFS on , introducing the convergence exponents and for digits and weighted products. It leverages conformal dynamics, pressure theory, and Cantor-type constructions to determine the Hausdorff dimension spectra of the corresponding level sets, showing that for , and revealing a piecewise spectrum for nondecreasing digits restricted to with a threshold at . Specifically, for , the spectrum is when , for , and at , with and sharing the same dimensional behavior. The work also connects these spectra to Lebesgue conformality and invariant measures, enriching the multifractal theory for infinite conformal systems and providing tools for analyzing digit-growth and restricted-growth sets in such systems.

Abstract

Let be the sequence of digits of in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence of the digits and the weighted products of distinct digits with finite numbers respectively, and then calculate the Hausdorff dimensions of the intersection of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are non-decreasing in such iterated function systems.
Paper Structure (13 sections, 14 theorems, 97 equations)

This paper contains 13 sections, 14 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. $d$-decaying Gauss like iterated function systems
  3. Confomal iterated function systems and relevant conclusions
  4. Statements of results
  5. Some useful lemmas
  6. Proofs of the main results
  7. Proof of Theorem \ref{['dg']}
  8. Proof of Theorem \ref{['ddg']}
  9. Proof of Theorem \ref{['jqdg']}
  10. Case $0\leq\alpha<\Sigma_t$
  11. Case $\Sigma_{t}\leq\alpha<\infty$
  12. Case $\alpha=\infty$
  13. Proof of Theorem \ref{['supp']}

Key Result

Theorem 1.1

(JR) Let $\Phi:\mathbb{N}\rightarrow\mathbb{R}$ be a function such that $n\leq\Phi(n)\leq\beta n$ for some $\beta\geq1$, then where we use $\dim_{\rm H}$ to denote the Hausdorff dimension.

Theorems & Definitions (20)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3: Mau95,MU96
  • Lemma 1.4: Mau95
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 10 more