Work

Shadowing for infinite dimensional Morse–Smale dynamical systems

Year: 2026

Type: article

Abstract: In this paper we extend to an infinite dimensional setting some results on the Shadowing property that are known on finite dimensional compact manifolds without border and in $$\mathbb {R}^n$$ R n . In fact, we show that if $$\{{\mathcal {T}}(t):t\ge 0\}$$ { T ( t ) : t ≥ 0 } is a Morse-Smale semigroup defined in a Hilbert space, with global attractor $$\mathcal {A}$$ A and non-wandering set given by its equilibria, then $${\mathcal {T}}(1)|_{\mathcal {A}}:\mathcal {A}\rightarrow \mathcal {A} $$ T ( 1 ) | A : A → A admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $$\mathcal {U}\supset \mathcal {A}$$ U ⊃ A of the global attractor, the map $${\mathcal {T}}(1)|_{\mathcal {U}}:\mathcal {U}\rightarrow \mathcal {U}$$ T ( 1 ) | U : U → U has the Hölder–Shadowing property. As applications, we obtain results related to the structural stability of Morse-Smale semigroups, that were only known on finite dimension, and continuity of global attractors.

Source: Mathematische Annalen

DOI: 10.1007/s00208-026-03409-5

Authors: José M. Arrieta, Alexandre N. Carvalho, Carlos R. Takaessu