Topological stability for functional dynamics

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In this paper, we deal with the functional envelope (S(X),Ff) of a homeomorphism f:X→X on a compact metric space X. We prove that if (S(X),Ff) has the shadowing property, then f also has the shadowing property, and the converse holds under the condition that f is expansive. We prove that every expansive functional envelope with the shadowing property is topologically stable in the class of all functional envelopes. Finally, we show that the topological stability is invariant under the concept of the functional envelopes.