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On the $L^q$ dimension of stationary measures for Möbius iterated function systems

Shunsuke Usuki

TL;DR

The paper advances the understanding of L^q dimensions for stationary measures of Möbius IFS under a strongly Diophantine condition by extending Shmerkin’s L^q framework to the SL_2(ℝ) action on RP^1. It establishes a dichotomy for the L^q spectrum τ(ν,q), tied to the canonical pressure zero \\widetilde{τ}(q), and proves a robust L^q norm flattening result via a novel L^q norm porosity argument that linearizes a non-linear G-action. It also exhibits counterexamples showing that a naïve extension of the linear case to Möbius IFS can fail, while revealing a rich second regime where τ(ν,q) grows linearly as αq beyond a threshold q0. The combination of uniform hyperbolicity, entropy- and pressure-based analysis, and a new porosity mechanism yields sharp control on the multifractal structure of stationary measures and clarifies when the L^q formalism aligns with the pressure predictions. Overall, the work connects dynamical systems, fractal geometry, and additive-combinatorics-inspired techniques to illuminate L^q regularity and its limitations for Möbius IFSs.

Abstract

We study the $L^q$ dimension $D(ν,q)\ (q>1)$ of stationary measures $ν$ for Möbius iterated function systems on $\mathbb{R}$ satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]{Shm19}. As the result, we show that there is the dichotomy: the $L^q$ spectrum $τ(ν,q)=(q-1)D(ν,q)$ is equal to the desired value $\min\{\widetildeτ(ν,q),q-1\}$ for any $q>1$, where $\widetildeτ(ν,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<α<1$ such that $τ(ν,q)=\min\{\widetildeτ(ν,q),q-1\}$ for $1<q<q_0$ and $τ(ν,q)=αq$ for $q\geq q_0$. In addition, we give examples of Möbius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.

On the $L^q$ dimension of stationary measures for Möbius iterated function systems

TL;DR

The paper advances the understanding of L^q dimensions for stationary measures of Möbius IFS under a strongly Diophantine condition by extending Shmerkin’s L^q framework to the SL_2(ℝ) action on RP^1. It establishes a dichotomy for the L^q spectrum τ(ν,q), tied to the canonical pressure zero \\widetilde{τ}(q), and proves a robust L^q norm flattening result via a novel L^q norm porosity argument that linearizes a non-linear G-action. It also exhibits counterexamples showing that a naïve extension of the linear case to Möbius IFS can fail, while revealing a rich second regime where τ(ν,q) grows linearly as αq beyond a threshold q0. The combination of uniform hyperbolicity, entropy- and pressure-based analysis, and a new porosity mechanism yields sharp control on the multifractal structure of stationary measures and clarifies when the L^q formalism aligns with the pressure predictions. Overall, the work connects dynamical systems, fractal geometry, and additive-combinatorics-inspired techniques to illuminate L^q regularity and its limitations for Möbius IFSs.

Abstract

We study the dimension of stationary measures for Möbius iterated function systems on satisfying the strongly Diophantine condition, and try the extension of Shmerkin's result \cite[Theorem 6.6]{Shm19}. As the result, we show that there is the dichotomy: the spectrum is equal to the desired value for any , where is the zero of the canonical pressure function, or there exist and such that for and for . In addition, we give examples of Möbius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.
Paper Structure (32 sections, 49 theorems, 640 equations, 1 figure)

This paper contains 32 sections, 49 theorems, 640 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on the dimension of stationary measures for IFSs
  3. Möbius IFSs and the main problem
  4. The main theorem
  5. About the proof: the $L^q$ norm flattening theorem
  6. About the proof: linearization and the $L^q$ norm porosity lemma
  7. Organization of the paper
  8. Notation
  9. Preliminaries
  10. Geometry of the ${\rm SL}_2(\mathbb{R})$-action on $\mathbb{RP}^1$
  11. The action of a uniformly hyperbolic family in ${\rm SL}_2(\mathbb{R})$
  12. Properties of stationary measures
  13. Effective separation of ${\rm SL}_2(\mathbb{R})$ by the action on $\mathbb{RP}^1$
  14. Multifractal properties of stationary measures
  15. Counterexamples to Problem \ref{['main_problem_L^q_dim_of_Furstenberg_measures']}
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\Phi=\{\varphi_i(x)=\lambda_ix+a_i\}_{i\in\mathcal{I}}$ be a linear IFS on $\mathbb{R}$ and assume that $\Phi$ satisfies the exponential separation condition along a subsequence. Then, if $p=(p_i)_{i\in\mathcal{I}}$ is a probability vector and we write $\nu$ for the stationary measure (self-sim where $H(p)=-\sum_{i\in\mathcal{I}}p_i\log p_i$ is the entropy of $p$ and $\chi(\Phi,p)=-\sum_{i\in

Figures (1)

  • Figure 1: $x_1,x_2,x_3$ mapped by $k$ and $a$

Theorems & Definitions (92)

  • Definition 1.1: Exponential separation condition along a subsequence for linear IFSs
  • Theorem 1.1: Hoc14
  • Definition 1.2
  • Theorem 1.2: HS17
  • Definition 1.3
  • Theorem 1.3: Shm19
  • Definition 1.4: Uniform hyperbolicity
  • Proposition 1.4: ABY10
  • Proposition 1.5
  • Theorem 1.7: Counterexamples to Problem \ref{['main_problem_L^q_dim_of_Furstenberg_measures']}
  • ...and 82 more