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

Abstract: We study the $L<sup>q$ dimension $D(\nu,q)\ (q&gt;1)$ 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 $L<sup>q$ spectrum $\tau(\nu,q)=(q-1)D(\nu,q)$ is equal to the desired value $\min{\widetilde{\tau}(\nu,q),q-1}$ for any $q&gt;1$, where $\widetilde{\tau}(\nu,q)$ is the zero of the canonical pressure function, or there exist $q_0&gt;1$ and $0&lt;\alpha&lt;1$ such that $\tau(\nu,q)=\min{\widetilde{\tau}(\nu,q),q-1}$ for $1&lt;q&lt;q_0$ 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}.