2019/08/02 by Alexander Condori, Condori, Alexander, Silas L. Carvalho +1
Mathematics · Physics and Astronomy · #28A78 #28D05 #37B10 #37B20 #37C05 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1908.00998
openalex publication_date 2019/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate in this work some situations where it is possible to estimate\nor determine the upper and the lower q-generalized fractal dimensions\nD\±\μ(q), q\∈\ℝ, of invariant measures associated with\ncontinuous transformations over compact metric spaces. In particular, we\npresent an alternative proof of Young's Theorem~ citeYoung for the\ngeneralized fractal dimensions of the Bowen-Margulis measure associated with a\nC1+\α-Axiom A system over a two-dimensional compact Riemannian\nmanifold M. We also present estimates for the generalized fractal dimensions\nof an ergodic measure for which Brin-Katok's Theorem is satisfied punctually,\nin terms of its metric entropy.\n Furthermore, for expansive homeomorphisms (like C1-Axiom A systems), we\nshow that the set of invariant measures such that D_\μ+(q)=0 (q\≥ 1),\nunder a hyperbolic metric, is generic (taking into account the weak topology).\nWe also show that for each s\∈ [0,1), D+\μ(s) is bounded above, up\nto a constant, by the topological entropy, also under a hyperbolic metric.\n Finally, we show that, for some dynamical systems, the metric entropy of an\ninvariant measure is typically zero, settling a conjecture posed by Sigmund\nin~ citeSigmund1974 for Lipschitz transformations which satisfy the\nspecification property.\n