On the support of measures of large entropy for polynomial-like maps
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917767430537216 |
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| author | Bazarbaev, Sardor Bianchi, Fabrizio Rakhimov, Karim |
| author_facet | Bazarbaev, Sardor Bianchi, Fabrizio Rakhimov, Karim |
| contents | Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $μ$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*δ_a$ towards $μ$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - a result due to de Thélin and Dinh - which does not rely on the existence of a Green current. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the support of measures of large entropy for polynomial-like maps Bazarbaev, Sardor Bianchi, Fabrizio Rakhimov, Karim Dynamical Systems Complex Variables Let $f$ be a polynomial-like map with dominant topological degree $d_t\geq 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $μ$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*δ_a$ towards $μ$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of $\mathbb P^k(\mathbb C)$ - a result due to de Thélin and Dinh - which does not rely on the existence of a Green current. |
| title | On the support of measures of large entropy for polynomial-like maps |
| topic | Dynamical Systems Complex Variables |
| url | https://arxiv.org/abs/2409.02039 |