On genericity of shadowing and periodic shadowing property

Abstract

Let M be a smooth compact manifold (possibly with boundary) or compact manifold with dim M 3 or dim M 6. We prove that generic homeomorphism on M has the periodic shadowing property.  2005 Elsevier Inc. All rights reserved.

On the genericity of the shadowing property for conservative homeomorphisms

“…• As mentioned briefly in the introduction, the proof actually still holds in the non-conservative case and provides an alternative proof to [9]. Indeed, topologically, the only difference is that the image of a cube may be strictly contained in another cube but this fact does not have any consequence on our proof.…”

“…B. Plamenevskaya were able to improve this result in [15] to any dimension in the case of smooth manifolds. In 2005, P. Koscielniak established in [9] the genericity of the shadowing property for homeomorphisms on a compact manifold which possesses a triangulation (smooth manifolds or topological manifolds of dimension ≤ 3 for example) or a handle decomposition (smooth manifolds or manifolds of dimension ≥ 6 for example). To the best of our knowledge, this was the best result obtained so far.…”

“…To the best of our knowledge, this was the best result obtained so far. Notice that our global strategy of proof is similar that of [9].…”

Generic properties of Z2-actions on the interval

“…Another important property is that a chain recurrent set is a Cantor set. In recent years these properties and some others have appeared to be C 0 -generic in the space of homeomorphisms on a compact manifold (see [1][2][3][4][5][6]8]). Meanwhile the concept of shadowing and of the chain recurrent set have been extended for dynamical system with multidimensional times, i.e., for Z d -actions [7,9].…”

Chaos and the shadowing property

“…This assumption was just relaxed by Kościelniak[13,15]. In his proof there is no need to use the topological transversality theorem any more, but covering relations are necessary as before.…”