A topological shadowing theorem

Topological stability for homeomorphisms with global attractor

“…The examples in this family are expansive with the shadowing property but not uniformly expansive. Then, we deduce their topological stability not from [22] (or [12]) but from Lemma 2.2. Now, recall that a continuous map of a metric space f ∶ X → X is dissipative if it possesses a compact absorbing set C; that is, for any bounded B ⊂ X there is n 0 (B) ∈ N ∪ {0} such that…”

“…The examples in this family are expansive with the shadowing property but not uniformly expansive. Then, we deduce their topological stability not from [22] (or [12]) but from Lemma 2.2.…”

“…The pursuit of extending this remarkable theorem to the challenging noncompact domain has spurred endeavors in multiple directions. Researchers have embarked on ambitious endeavors to bridge the gap between compact and noncompact settings, with notable contributions documented in a body of literature that includes works such as [5, 11, 12, 18]. These exploratory undertakings strive to generalize the Walters stability theorem and unearth the underlying principles governing the stability of dynamical systems in an increasingly diverse and complex landscape.…”