Abstract
For an α -expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the α -shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.
Highlights
In [1] (Theorem 1.2.1) it is proved, among other things, that Anosov diffeomorphisms has the shadowing property, called pseudo orbit tracing property there
On [1] (p. 23), the authors only uses the so-called local product structure property: if d( x, y) < δ Wεs ( x ) ∩ Wεu (y) 6= ∅ if δ > 0 is chosen small enough for a given ε > 0, and the special properties of the metric d coming from the Riemannian structure of the manifold supporting the system [1] ((B), p. 20)
As can be checked the first of these two conditions is equivalent to the shadowing property for pseudo orbits with one jump, that is, for every ε > 0 there exists δ > 0 such that for every bi-sequence of points of the form
Summary
In [1] (Theorem 1.2.1) it is proved, among other things, that Anosov diffeomorphisms has the shadowing property, called pseudo orbit tracing property there. In [2] (Theorem 5.1) it is shown that for every expansive homeomorphism on a compact space there exists a compatible metric (which we call hyperbolic metric) with similar properties to those of the metric d in the case of Anosov diffeomorphisms. The proof of the shadowing property for Anosov diffeomorphisms given in [1] carries over the more general case of expansive systems. We give an alternative and elementary proof of this well-known shadowing condition (Proposition 4), not making use of Fathi’s hyperbolic metric. Instead we use a reformulation of the property of expansiveness of a system (Proposition 1) which seems interesting in its own right
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