On the $L^q$ dimension of stationary measures for M\"{o}bius iterated function systems

TL;DR

This paper investigates the $L^q$ dimension of stationary measures for Möbius IFS on the real line, revealing a dichotomy in the $L^q$ spectrum and providing examples that illustrate this behavior.

Contribution

It extends Shmerkin's results to Möbius IFS under Diophantine conditions, establishing a dichotomy in the $L^q$ spectrum and answering Solomyak's question positively.

Findings

The $L^q$ spectrum exhibits a dichotomy based on the zero of the pressure function.

Explicit examples demonstrate the case where the spectrum is linear for large $q$.

The results generalize previous work to a new class of IFS with Möbius transformations.

Abstract

We study the $L^q$ dimension $D(\nu,q)\ (q>1)$ of stationary measures $\nu$ for M\"{o}bius 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 $\tau(\nu,q)=(q-1)D(\nu,q)$ is equal to the desired value $\min\{\widetilde{\tau}(\nu,q),q-1\}$ for any $q>1$, where $\widetilde{\tau}(\nu,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<\alpha<1$ such that $\tau(\nu,q)=\min\{\widetilde{\tau}(\nu,q),q-1\}$ for $1<q<q_0$ and $\tau(\nu,q)=\alpha q$ for $q\geq q_0$. In addition, we give examples of M\"{o}bius iterated function systems which show the latter case by giving an affirmative answer to Solomyak's question \cite[Question 2]{Sol24}.