\( C^1\)-stable shadowing diffeomorphisms. (English) Zbl 1153.37316

Lee, Keonhee; Moriyasu, Kazumine; Sakai, Kazuhiro

Discrete Contin. Dyn. Syst. 22, No. 3, 683-697 (2008).

Summary: Let \(f\) be a diffeomorphism of a closed \(C^\infty\) manifold. In this paper, we define the notion of the \(C^1\)-stable shadowing property for a closed \(f\)-invariant set, and prove that (i) the chain recurrent set \(\mathcal R(f)\) of \(f\) has the \(C^1\)-stable shadowing property if and only if f satisfies both Axiom A and the no-cycle condition, and (ii) for the chain component \(C_f(p)\) of \(f\) containing a hyperbolic periodic point \(p, C_f ( p)\) has the \(C^1\)-stable shadowing property if and only if \(C_f(p)\) is the hyperbolic homoclinic class of \(p\).

MSC:

37B20	Notions of recurrence and recurrent behavior in topological dynamical systems
37C29	Homoclinic and heteroclinic orbits for dynamical systems
37C50	Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37D30	Partially hyperbolic systems and dominated splittings
37C75	Stability theory for smooth dynamical systems

Keywords:

\(C^1\)-stable shadowing property; pseudo-orbit; shadowing; chain recurrent set; chain component; homoclinic class; hyperbolic set; axiom A