Dimension of Furstenberg measures on $\mathbb{CP}^{1}$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908643414245376 |
|---|---|
| author | Rapaport, Ariel Ren, Haojie |
| author_facet | Rapaport, Ariel Ren, Haojie |
| contents | Let $θ$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(θ)$ is strongly irreducible and proximal. Let $μ$ denote the Furstenberg measure on $\mathbb{CP}^{1}$ associated to $θ$. Assume further that no generalized circle is fixed by all Möbius transformations corresponding to elements of $\mathcal{G}$, and that $\mathcal{G}$ satisfies a mild Diophantine condition. Under these assumptions, we prove that $\dimμ=\min\left\{ 2,h_{\mathrm{RW}}/\left(2χ\right)\right\} $, where $h_{\mathrm{RW}}$ and $χ$ denote the random walk entropy and Lyapunov exponent associated to $θ$, respectively.
Since our result expresses $\dimμ$ in terms of the random walk entropy rather than the Furstenberg entropy, and relies only on a mild Diophantine condition as a separation assumption, we are forced to directly confront difficulties arising from the ambient space $\mathbb{CP}^{1}$ having real dimension $2$ rather than $1$. Moreover, our analysis takes place in a projective, contracting-on-average setting. This combination of features introduces significant challenges and requires genuinely new ideas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00729 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension of Furstenberg measures on $\mathbb{CP}^{1}$ Rapaport, Ariel Ren, Haojie Dynamical Systems 28A80, 37C45 Let $θ$ be a finitely supported probability measure on $\mathrm{SL}(2,\mathbb{C})$, and suppose that the semigroup generated by $\mathcal{G}:=\mathrm{supp}(θ)$ is strongly irreducible and proximal. Let $μ$ denote the Furstenberg measure on $\mathbb{CP}^{1}$ associated to $θ$. Assume further that no generalized circle is fixed by all Möbius transformations corresponding to elements of $\mathcal{G}$, and that $\mathcal{G}$ satisfies a mild Diophantine condition. Under these assumptions, we prove that $\dimμ=\min\left\{ 2,h_{\mathrm{RW}}/\left(2χ\right)\right\} $, where $h_{\mathrm{RW}}$ and $χ$ denote the random walk entropy and Lyapunov exponent associated to $θ$, respectively. Since our result expresses $\dimμ$ in terms of the random walk entropy rather than the Furstenberg entropy, and relies only on a mild Diophantine condition as a separation assumption, we are forced to directly confront difficulties arising from the ambient space $\mathbb{CP}^{1}$ having real dimension $2$ rather than $1$. Moreover, our analysis takes place in a projective, contracting-on-average setting. This combination of features introduces significant challenges and requires genuinely new ideas. |
| title | Dimension of Furstenberg measures on $\mathbb{CP}^{1}$ |
| topic | Dynamical Systems 28A80, 37C45 |
| url | https://arxiv.org/abs/2511.00729 |