Lipschitz and Hölder Shadowing and Structural Stability

Abstract

In this chapter, we give either complete proofs or schemes of proof of the following main results:

• If a diffeomorphism f of a smooth closed manifold has the Lipschitz shadowing property, then f is structurally stable (Theorem 2.3.1);

• a diffeomorphism f has the Lipschitz periodic shadowing property if and only if f is Ω-stable (Theorem 2.4.1);

• if a diffeomorphism f of class C ² has the Hölder shadowing property on finite intervals with constants $\mathcal{L},C,d_{0},\theta,\omega$, where θ ∈ (1∕2, 1) and θ + ω > 1, then f is structurally stable (Theorem 2.5.1);

• there exists a homeomorphism of the interval that has the Lipschitz shadowing property and a nonisolated fixed point (Theorem 2.6.1);

• if a vector field X has the Lipschitz shadowing property, then X is structurally stable (Theorem 2.7.1).