On the generalized dimensions of chaotic attractors

Théophile Caby, Michele Gianfelice

Published 2023-09-14Version 1

We prove that if $\mu$ is the physical measure of a chaotic flow diffeomorphically conjugated to a suspension flow based on a Poincar\'{e} application $R$ with physical measure $\mu_{R}$, $D_{q}(\mu)=D_{q}(\mu _{R})+1$, where $D_{q}$ denotes the generalized dimension of order $q$ with $q \neq1$. We also prove that a similar result holds for the local dimensions so that, under the additional hypothesis of exact-dimensionality of $\mu_{R}$, our result extends to the case $q=1$. We then apply these results to estimate the $D_{q}$ spectrum of the R\"ossler attractor and prove the existence of the information dimension $D_{1}$ for the Lorenz '63 flow.