On the generalized dimensions of physical measures of chaotic flows

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.