On the generalized dimensions of physical measures of chaotic flows

• Published in: Chaos, solitons and fractals Vol. 199; p. 116678
• Main Authors: Caby, Théophile, Gianfelice, Michele
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.10.2025; Elsevier
• Subjects: Chaotic attractor; Generalized dimensions; Mathematics; Nonlinear Sciences; Rössler flow; Singular-hyperbolic attractors; Singular-hyperbolic attractors; 37C10; Chaotic attractor; 37D45; Generalized dimensions; 37E05; Rössler flow; Lorenz flow; singular-hyperbolic attractors; AMS subject classification: 37C10; AMS subject classification: 37C10 37D45 37E05 generalized dimensions chaotic attractor Rössler flow singular-hyperbolic attractors; 37E05 generalized dimensions; chaotic attractor
• ISSN: 0960-0779; 1873-2887; 1873-2887
• DOI: 10.1016/j.chaos.2025.116678

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. •We relate the dimension spectrum of a smooth flow to that of its first return map.•We apply our results to singular hyperbolic flows and bound their dimensions.•We compute the generalized dimensions of semi-flows modeling Rossler systems.