N-expansive homeomorphisms on surfaces

Abstract

We exploit the techniques developed in [Le] to study N -expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M with nonwandering set Ω(f ) being the whole of M then f is expansive. This condition on the nonwandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on a surface with genus 2 with wandering points that is not expansive. * M. J.

Dendritations of surfaces

“…This is the leaves-atlas equivalence mentioned in the beginning of this section. The next example shows that a maximal atlas (with non-locally connected plaques) can contain strictly more information than the leaves and the basis, i.e., does not satisfy (6).…”

“…We have that each A n has at least three points. We can take a n , b n ∈ A n such that dist(a n , b n ) → 0 and y n ∈ Q(a n , b n ) 6 . This contradicts that Q is C-smooth and proves that the set of ramification points can only accumulate on ∂D.…”

“…Take a plane flow (with X a compact invariant annulus) with limit cycles. An orbit converging to a cycle is locally connected but it is not compact.Note that the examples given above (Examples 4.2.3 and 4.2.4) are not Peano foliations but they satisfy(6) in Theorem 4.2.2. The next subsection is devoted to describe an example of a C-foliation not satisfying(6).…”

General Expansiveness for Diffeomorphisms from the Robust and Generic Properties

“…In fact, if 1-expansive then it is expansive. Note that Artigue et al [2] proved that if every 2-expansive homeomorphism without wandering points of a compact boudaryless surface is expansive. Moreover, they exhibit a 2-expansive homeomorphism on the two torus whose nonwandering set is a proper subset of two torus that is not expansive.…”

“…Moreover, they exhibit a 2-expansive homeomorphism on the two torus whose nonwandering set is a proper subset of two torus that is not expansive. Very recently, Artigue [2] studied robustly n-expansive and showed that for each r ≥ 2 there is a C r (r ≥ 2)-robustly r-expansive surface diffeomorphism that is not (r − 1)-expansive. Li and Zhang [11] given an example that there is a n + 1 expansive homeomorphism on a compact metric spaces such that it is not a n-expansive homeomorphism for all n ≥ 1.…”