Measurability of Multifractal Topological Entropy and Its Role in Multifractal Theory
Tingting Wang, Bilel Selmi, Zhiming Li
TL;DR
This work develops a dynamical multifractal framework by introducing $(q, \vartheta)$-Bowen and $(q, \vartheta)$-packing topological entropies to quantify local entropy structures. It establishes measurability and descriptive-set-theoretic regularity of these entropy maps, relates the entropy on level sets $L_{\beta}$ to the packing entropy via $\mathscr{E}_{top}^{P}(f, L_{\beta}) = q\beta + \mathscr{E}_{\vartheta, q}^{P}(f, L_{\beta})$ under suitable conditions, and characterizes the domain of the multifractal spectrum through Legendre transforms $h^{*}(\beta)$. The domain results show $L_{\beta}$ is nonempty only for $\beta$ within a computable interval $[\underline{\beta}, \overline{\beta}]$, with equality $\mathscr{E}_{top}^{P}(f, L_{\beta}) = h^{*}(\beta)$ under regularity. Two key examples—Anosov toral automorphisms and Gibbs measures for Anosov diffeomorphisms—illustrate trivial and nontrivial spectral behaviour and yield explicit Legendre-transform representations of the multifractal spectra.
Abstract
In this paper, we consider definitions including $(q, \vartheta)$-Bowen topological entropy and $(q, \vartheta)$-packing topological entropy. We systematically explore their properties and measurability and analyze the relationship between $(q, \vartheta)$-packing topological entropy and topological entropy on level sets. Furthermore, the study demonstrates that the domain of $(q, \vartheta)$-packing topological entropy encompasses the domain of the multifractal spectrum of local entropies, providing new perspectives and tools for multifractal analysis.