On the generalized dimensions of physical measures of chaotic flows

We prove that if μ is the physical measure of a C2 flow in Rd,d≥3, diffeomorphically conjugated to a suspension flow based on a Poincaré application R with physical measure μR, then Dq(μ)=Dq(μR)+1, where Dq denotes the generalized dimension of order q≠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 μ and, under the additional hypothesis of exact-dimensionality of μR, that our result extends to the case q=1. We apply these results to estimate the Dq 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.