On the $L^q$ dimension of stationary measures for Möbius iterated function systems
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912201458057216 |
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| author | Usuki, Shunsuke |
| author_facet | Usuki, Shunsuke |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $L^q$ dimension of stationary measures for Möbius iterated function systems Usuki, Shunsuke Dynamical Systems Classical Analysis and ODEs Combinatorics 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}. |
| title | On the $L^q$ dimension of stationary measures for Möbius iterated function systems |
| topic | Dynamical Systems Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2501.13729 |