Article Dans Une Revue Chaos, Solitons & Fractals Année : 2025

On the generalized dimensions of physical measures of chaotic flows

Résumé

We prove that if $\mu$ 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 Poincar\'{e} application $R$ with physical measure $\mu_{R}$, then $D_{q}(\mu)=D_{q}(\mu _{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q \neq1$. We also show that a similar result holds for the local dimensions of $\mu$ and, under the additional hypothesis of exact-dimensionality of $\mu_{R}$, that our result extends to the case $q=1$. We apply these results to estimate the $D_{q}$ spectrum associated with R\"ossler 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.

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hal-04541549 , version 1 (10-04-2024)

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Théophile Caby, Michele Gianfelice. On the generalized dimensions of physical measures of chaotic flows. Chaos, Solitons & Fractals, 2025, 199, pp.116678. ⟨10.1016/j.chaos.2025.116678⟩. ⟨hal-04541549⟩
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