On the generalized dimensions of physical measures of chaotic flows

We prove that if $\mu$ is the physical measure of a $C^2$ flow in ${\mathbb{R}}^d,d\ge 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré 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 $q\mathrm{\ne }1$. 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 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.