On the generalized dimensions of physical measures of chaotic flows
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912691945209856 |
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| author | Caby, Théophile Gianfelice, Michele |
| author_facet | Caby, Théophile Gianfelice, Michele |
| contents | We prove that if $μ$ is the physical measure of a $C^2$ flow in $\mathbb{R}^d, d \geq 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré application $R$ with physical measure $μ_{R}$, then $D_{q}(μ)=D_{q}(μ_{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q \neq1$. We also show that a similar result holds for the local dimensions of $μ$ and, under the additional hypothesis of exact-dimensionality of $μ_{R}$, that our result extends to the case $q=1$. We apply these results to estimate the $D_{q}$ spectrum associated with Rössler systems and turn our attention to Lorenz-like flows, proving the existence of their information dimension and giving a lower bound for their generalized dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07575 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the generalized dimensions of physical measures of chaotic flows Caby, Théophile Gianfelice, Michele Dynamical Systems 37C10, 37D45, 37E05 We prove that if $μ$ is the physical measure of a $C^2$ flow in $\mathbb{R}^d, d \geq 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré application $R$ with physical measure $μ_{R}$, then $D_{q}(μ)=D_{q}(μ_{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q \neq1$. We also show that a similar result holds for the local dimensions of $μ$ and, under the additional hypothesis of exact-dimensionality of $μ_{R}$, that our result extends to the case $q=1$. We apply these results to estimate the $D_{q}$ spectrum associated with Rössler systems and turn our attention to Lorenz-like flows, proving the existence of their information dimension and giving a lower bound for their generalized dimensions. |
| title | On the generalized dimensions of physical measures of chaotic flows |
| topic | Dynamical Systems 37C10, 37D45, 37E05 |
| url | https://arxiv.org/abs/2309.07575 |