Stable and unstable sets ofC⁰perturbations of expansive homeomorphisms of surfaces

Abstract

Let M be a compact metric space and g : M → M be an homeomorphism C 0 -close to an expansive map of M. In general, it is not true that g is also expansive, but it still has some properties resembling the expansivity. In fact, if we identify pairs of points whose g-orbits stay nearby, both for the future and the past, we obtain an equivalence relation ∼. The quotient space M/ ∼ is a compact, metric space and g induces an expansive homeomorphism g on that quotient. If M is a surface, we show that for any x ∈ M/ ∼ the connected component of the local stable (unstable) set containing x is nontrivial and arc-wise connected.

Extensions of expansive dynamical systems

“…To see that q g ∈ N q pick ε > 0 such that B ε (q) ⊆ N q , and let N ∈ N as in condition (5). Taking a smaller neighborhood N f if necessary we can suppose that…”

“…and studied the properties of the quotient space M/∼ and the quotient homeomorphismf : M/∼ → M/∼, suggesting thatf is expansive and showing that if f 0 is persistent then there is a copy of (M, f 0 ) inside (M/∼,f ). In a subsequent paper, [5], Cerminara and Sambarino proved in full detail thatf is expansive, and for the case of M a surface they showed that the local stable and unstable sets of the system (M/∼,f ) are non-trivial and arc-wise connected. The aim of this paper is to study the previous situation assuming directly a property of the form (1) for f without supposing the existence of f 0 expansive; in fact the existence of such an f 0 now turns to be a question.…”

“…In [8, p. 568] it is suggested that what we called Lewowicz quotient (Definition 3.3) in honor of the author of that article is in fact an expansive homeomorphism. Full details and proofs of this assertion was given later in [5], where the authors construct a metric in the quotient space with respect to which the quotient homeomorphism is expansive. In the following result we give an alternative proof of this fact with a shorter but non constructive argument.…”

New cw-expansive homeomorphisms of surfaces

“…In [16] it is claimed that in this case the quotient space M = M/ ∼ is metrizable and the induced homeomorphism f : M → M is expansive. The details of this construction were given in [8] by Cerminara and Sambarino.…”

“…for every continuum C ⊆ M . In [11] the techniques of [8] were applied to this case, proving that the corresponding quotient is cw-expansive.…”