C^1-stable inverse shadowing chain components for generic diffeomorphisms
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Let f be a diffeomorphism of a compact C^∞ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of C^1-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set R(f) of f has C^1-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) C^1-generically, the chain component C_f(p) of f associated to p is hyperbolic if and only if C_f(p) has the C^1-stable inverse shadowing property.
Abstract
Let f be a diffeomorphism of a compact C^∞ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of C^1-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set R(f) of f has C^1-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) C^1-generically, the chain component C_f(p) of f associated to p is hyperbolic if and only if C_f(p) has the C^1-stable inverse shadowing property.
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