On the generalized dimensions of physical measures of chaotic flows

TL;DR

This paper establishes a relationship between the generalized dimensions of physical measures of certain chaotic flows and their Poincaré maps, providing new insights into the fractal structure of these measures.

Contribution

It proves a formula relating the generalized dimensions of physical measures of $C^2$ flows to those of their Poincaré maps, extending to local and information dimensions under specific conditions.

Findings

Derived a formula for $D_q$ of flow measures based on Poincaré maps.

Applied results to R"ossler systems to estimate their spectra.

Proved the existence of information dimension for Lorenz-like flows and provided lower bounds for their generalized dimensions.

Abstract

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.