Different Statistical Behaviors of Orbits

TL;DR

Problem: quantify how long-run orbit statistics influence global dynamical complexity, especially for nonrecurrent points. Approach: introduce xi-$\omega$-limit sets, a 56-pattern taxonomy of statistical futures, and saturated/entropy-dense methods; prove nonrecurrent points carry full $h_{top}$ under shadowing/almost specification; demonstrate existence of 50 realizable cases and detail entropy and measure-theoretic properties across these classes. Key contributions: (i) basic characterizations linking $\omega$-limit structures to empirical measures; (ii) a comprehensive 50-case landscape for orbit statistics; (iii) entropy estimates showing many patterns realize full entropy in transitive expanding or Anosov contexts; (iv) Lebesgue-measure and dynamical-differentiatied observations across beta-shifts, $C^{1+\alpha}$ surface diffeomorphisms, and Mañé diffeomorphisms; (v) concrete corollaries for classical models via beta-shifts and smooth dynamical systems. Significance: unifies topological and statistical perspectives on orbit futures, enabling precise entropy and measure-theoretic conclusions for symbolic, smooth, and flow-based systems with broad applicability.

Abstract

In this paper, we will study the statistical behaviors of orbits. Firstly, we will show that for a dynamical systems have the shadowing property or almost specification property, the set of nonrecurrent points has full topological entropy. After that, we introduce a criteria for classification of dynamical orbits in order to study the complexity theory of dynamical systems. The criteria is to use upper and lower natural density, upper and lower Banach density to divide different statistical future of dynamical orbits into 56 cases, 28 cases for recurrent orbits and 28 cases for nonrecurrent orbits. We will show the existence of 50 cases and for topologically transitive topologically expanding or topologically transitive topologically Anosov dynamical systems, we will prove that 35 classes, including all the 28 cases for nonrecurrent orbits, can carry full topological entropy. Besides, we will prove that 9 cases can be observable in some differential dynamical systems. Finally, we will apply our results to $\b{eta}-$shifts, $C^{1+α}$ surface diffeomorphisms and Ma$ñ\'$e diffeomorphisms.

Figures (2)

• Figure 1: Bowen’s eye
• Figure 2: The figure-eight attractor

Theorems & Definitions (130)

• Theorem A
• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Theorem B
• Remark 1.5
• Definition 1.6
• Remark 1.7
• Remark 1.8