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.