Shadowing for infinite dimensional Morse–Smale dynamical systems

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 $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . In fact, we show that if $$\{{\mathcal {T}}(t):t\ge 0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>T</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a Morse-Smale semigroup defined in a Hilbert space, with global attractor $$\mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> and non-wandering set given by its equilibria, then $${\mathcal {T}}(1)|_{\mathcal {A}}:\mathcal {A}\rightarrow \mathcal {A} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> admits the Lipschitz Shadowing property. Moreover, for any positively invariant bounded neighborhood $$\mathcal {U}\supset \mathcal {A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>⊃</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> of the global attractor, the map $${\mathcal {T}}(1)|_{\mathcal {U}}:\mathcal {U}\rightarrow \mathcal {U}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mi>U</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mi>U</mml:mi> <mml:mo>→</mml:mo> <mml:mi>U</mml:mi> </mml:mrow> </mml:math> 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.