Different Asymptotic Behavior versus Same Dynamical Complexity

Abstract

For any dynamical system $T:X\to X$ of a compact metric space $X$ with $g-$almost product property and uniform separation property, under the assumptions that the periodic points are dense in $X$ and the periodic measures are dense in the space of invariant measures, we distinguish various periodic-like recurrences and find that they all carry full topological topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multi-fractal analysis for all continuous observable functions. These results are suitable for all $\beta -$shfits ($\beta >1$), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc. Roughly speaking, we combine many different "eyes" (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a {\it Refined Dynamical Structure} for Recurrence Theory and Multi-fractal Analysis.

61 pages. For a certain class of dynamical systems such as $\beta$ shifts, mixing subshifts of finite type and mixing hyperblic systems, we study various gap-sets of periodic-like recurrence and (ir)regularity and show that they all carry full topological entropy

References in corpus (3)

• Topological pressure of simultaneous level sets
• Symbolic extensions for partially hyperbolic diffeomorphisms
• The Structure on Invariant Measures of $C^1$ generic diffeomorphisms

Cited by in corpus (2)

• The upper capacity topological entropy of free semigroup actions for certain non-compact sets,$II$
• On the topological entropy of saturated sets for amenable group actions