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

Abstract

We study the $L^q$ dimension $D(\nu ,q)(q>1)$ of stationary measures $\nu$ 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 $\tau (\nu ,q)=(q-1)D(\nu ,q)$ is equal to the desired value $\min \{\tilde{\tau }(\nu ,q),q-1\}$ for any $q>1$, where $\tilde{\tau }(\nu ,q)$ is the zero of the canonical pressure function, or there exist $q_0>1$ and $0<\alpha <1$ such that $\tau (\nu ,q)=\min \{\tilde{\tau }(\nu ,q),q-1\}$ for $1<q<q_0$ and $\tau (\nu ,q)=\alpha q$ for $q\ge q_0$. 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}.

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