Persistent Shadowing in Functional Dynamics and Its Measure-Theoretic Implications

Document Type : Original Article

Author

Ali Barzanouni

Department of Mathematics and computer science, Hakim Sabzevari university

10.22034/kjm.2025.472943.3283

Abstract

We introduce the persistent shadowing property for continuous group actions $\varphi: G \times X \to X$, establishing its measure-theoretic foundations through compatible Borel probability measures $\mu \in \mathcal{M}_{PSh}(X, \varphi)$. We prove $\mathcal{M}_{PSh}(X, \varphi)$ is an $F_{\sigma \delta}$-set in $\mathcal{M}(X)$, and show $\varphi$ has persistent shadowing on $\text{supp}(\mu)$ when $\mu \in \mathcal{M}_{PSh}(X, \varphi)$. For compact $X$ without isolated points, we characterize when all non-atomic measures imply persistent shadowing.

For functional envelopes $\tilde{\varphi}: G \times S(X) \to S(X)$, we demonstrate that topological stability and expansivity are preserved, yielding topological stability for expansive envelopes with weak shadowing. We establish that shadowing properties of $\tilde{\varphi}$ descend to $\varphi$, with converses holding when $\varphi$ is expansive.

Keywords

Functional envelope

topologically stable

shadowing

persistent

Borel measure

Subjects

• 37 Dynamical systems and ergodic theory

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