A Note on the Relation Between the Metric Entropy and the Generalized Fractal Dimensions of Invariant Measures

Condori A.; Carvalho S.L. · Bulletin of the Brazilian Mathematical Society (2022) · DOI: 10.1007/s00574-021-00266-5

Resumen

We investigate in this work some situations where it is possible to estimate or determine the upper and the lower q-generalized fractal dimensions Dμ±(q), q∈ R, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young’s Theorem by Young (Ergod. Theory Dyn. Syst. 2(1):109–124, 1982) for the generalized fractal dimensions of the Bowen-Margulis measure associated with a C1+α-Axiom A system over a two-dimensional compact Riemannian manifold M. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok’s Theorem is pointwise satisfied, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like C1-Axiom A systems), we show that the set of invariant measures such that Dμ+(q)=0 (q≥ 1), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each s∈ [0 , 1) , Dμ+(s) is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund (Trans. Am. Math. Soc. 190:285–299, 1974) for Lipschitz transformations which satisfy the specification property.

Instituciones: Universidad Nacional de San Cristóbal de Huamanga