Transitively-Saturated Property, Banach Recurrence and Lyapunov Regularity

TL;DR

This paper introduces new levels of Banach recurrence, demonstrating they possess full topological entropy, and combines this with Lyapunov regularity to advance mixed multifractal analysis, extending existing saturation concepts.

Contribution

It presents five new levels of Banach recurrence with full entropy and refines saturation properties to transitively-saturated and transitively-convex-saturated, enhancing multifractal analysis.

Findings

Five new levels of Banach recurrence with full topological entropy.

Extension of saturation properties to transitively-saturated and transitively-convex-saturated.

Refined multifractal analysis combining Banach recurrence and Lyapunov regularity.

Abstract

The topological entropy of various gap-sets on periodic-like recurrence and Birkhoff regularity were considered in [69] but some Banach recurrence and Lyapunov regularity are not considered. In this paper we introduce five new levels on Banach recurrence and show they all carry full topological entropy, and simultaneously combine with Lyapunov regularity to get some refined theory on mixed multifractal analysis of [8,29]. In this process, we strengthen Pfister and Sullivan's result of [58] from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property).