Abstract

Firstly, we introduce the concept of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space. Secondly, we study their dynamical properties and obtain the following results. (1) If the map f has the G -shadowing property, then the map f is G -chain mixed if and only if the map f is G -mixed. (2) The map f is G -chain transitive if and only if for any positive integer k ≥ 2 , the map f k is G -chain transitive. (3) If the map f is G -pointwise chain recurrent, then the map f is G -chain transitive. (4) If there exists a nonempty open set U satisfying G U = U , U ¯ ≠ X , and f U ¯ ⊂ U , then we have that the map f is not G -chain transitive. These conclusions enrich the theory of G -chain mixing, G -mixing, and G -chain transitivity in metric G -space.

Highlights

  • Chain mixing, mixing, and chain transitivity are very important concepts in topological dynamical systems

  • (4) If there exists a nonempty open set U satisfying GðUÞ = U, U ≠ X, and f ðU Þ ⊂ U, we have that the map f is not G-chain transitive

  • It provided the theoretical basis and scientific foundation for the application of G-chain mixing, G-mixing, and G-chain transitivity in computational mathematics and biological mathematics

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Summary

Introduction

Chain mixing, mixing, and chain transitivity are very important concepts in topological dynamical systems. Many scholars studied their dynamical properties and obtained some valuable results (see [1,2,3,4,5,6,7,8,9]). We introduce the concept of G-chain mixing, G-mixing, and G-chain transitivity in metric G-space We study their dynamical properties and obtain the following theorem. Let ðX, dÞ be a compact metric G-space, f : X ⟶ X be a pseudo equivalent map, and topological group G be compact.

G-Chain Mixing and G-Mixing in Metric G-Space
G-Chain Transitive in Metric G-Space
Conclusion
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