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This chapter presents some basic elements of the theory of semigroup actions on topological spaces that appear in the study of Lyapunov stability. It presents the formal definition of semigroup action and the concepts of orbits, invariant sets, limit sets, prolongations, and prolongational limit sets for semigroup actions on topological spaces. It also presents some topological properties of uniform spaces, which are the topological spaces considered as phase space for the semigroup actions.

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Semigroup Actions

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  • 发表日期：

    2025年

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  • Victor H. L. Rocha,
  • Josiney A. Souza

摘要

This chapter presents some basic elements of the theory of semigroup actions on topological spaces that appear in the study of Lyapunov stability. It presents the formal definition of semigroup action and the concepts of orbits, invariant sets, limit sets, prolongations, and prolongational limit sets for semigroup actions on topological spaces. It also presents some topological properties of uniform spaces, which are the topological spaces considered as phase space for the semigroup actions.