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 $R^d,d\ge 3,$ diffeomorphically conjugated to a suspension flow based on a Poincaré application $R$ with physical measure ${\mu }_R$, then $D_q\left(\mu \right)=D_q\left({\mu }_R\right)+1$, where $D_q$ denotes the generalized dimension of order $q\ne 1$. The proof is different from those presented in [BSau] and [PS] for uniformly hyperbolic flows, therefore it extends this result also to the case of flows generated by three-dimensional vector fields having a global singular hyperbolic attractor ([AP], [AMe]). 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.