Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity

Abstract

For any dynamical system T : X → 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 β−shfits (β > 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 Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

Distributional chaos in multifractal analysis, recurrence and transitivity

“…For φ being an arbitrary continuous function (hence there may exist more than one equilibrium measures), (1.1) was established by Takens and Verbitski [66] under the assumption that f has the specification property. This result was further generalized by Pfister and Sullivan [54] to dynamical systems with g-product property(see [68,70] for more related discussions). The method used in [6,8] mainly depends on thermodynamic formalism such as differentiability of pressure function while the method in [66,54] is a direct approach by constructing fractal sets.…”

“…One can construct corresponding uncountable DC1-scrambled subset one by one but everyone needs a long construction proof so that it is not a good choice to do these constructions directly. Recall that in the case of entropy estimate on recurrent levels, one main technique chosen in [70,35] is using (transitively) saturated property which can avoid to do a long construction proof for every considered object. So here we follow the way of [70,35] to give a DC1 result in saturated sets.…”

Transitively-saturated property, Banach recurrence and Lyapunov regularity

“…In [69] R and QR d \R = QR d \QR erg were considered but QR erg \R = QR erg \QR d and QR \ (QR erg ∪ QR d ) are not considered.…”

“…Proof. All constructed K in [69] satisfy C K = X so that G K ⊆ QW ∩ T ran (in this case G K = G T K ). Thus transitively-saturated property is not necessary for this result and saturated property is enough.…”

“…Many periodic-like recurrence such as periodic, almost periodic, weakly almost periodic, quasi-weakly almost periodic and Birkhoff regularity were considered in [69] in the viewpoint of topological entropy. But some Banach recurrence and Lyapunov regularity are not considered.…”