Transitivity in CR-Dynamical Systems

TL;DR

The paper adapts topological dynamics to CR-dynamical systems by introducing transitivity trees and a fourth transitive-point type ($0$-transitive), along with two additional transitivity variants and a mixing notion. It develops a deep structural framework—transitivity trees, degenerate/nondegenerate points, and the equivalence relation $\sim_G$—to relate trajectories, Mahavier products, and forward/backward branches. It then analyzes $0$-, $1$-, $2$-, and $3$-transitive points and dense orbit transitivity, establishing density, $G_\delta$-classifications, and interdependencies among transitivity levels, including counterexamples that separate certain implications. The results connect SV-dynamics with classical transitivity, yielding a robust hierarchy of transitivity notions and providing criteria under which these notions are dense or generic, with explicit examples illustrating the sharp boundaries of various implications.

Abstract

A CR-dynamical system is a pair $(X, G)$, where $X$ is a compact metric space and $G$ is a closed relation (CR) on $X$. In this paper, we introduce a new type of transitive point and transitivity in CR-dynamical systems. We develop a new tool called transitivity trees, which we use to determine the relationship between the different types of transitive points.

Figures (14)

• Figure 1: Levels in a tree
• Figure 2: An infinite tree with no infinite branches
• Figure 3: Types of transitivity trees
• Figure 4: The Tent Map
• Figure 5: Transitivity tree $T_G{\left( {x} \right)}$ in Example \ref{['ex:1-transitive-not-0-transitive']}

Theorems & Definitions (149)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.7
• Definition 2.9: Definition $2.5$ transitivity_CR
• Definition 2.10
• Definition 2.11
• Definition 2.16