On the generalized dimensions of physical measures of chaotic flows

Théophile Caby and Michele Gianfelice

Chaos, Solitons & Fractals, 2025, vol. 199, issue P2

Abstract: 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.

Keywords: Generalized dimensions; Chaotic attractor; Rössler flow; Singular-hyperbolic attractors (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:199:y:2025:i:p2:s0960077925006915

DOI: 10.1016/j.chaos.2025.116678

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