We study dynamical systems for which at most $n$ orbits can accompany
a given arbitrary orbit.
For simplicity we call them $n$-expansive
(or positively $n$-expansive if positive orbits are considered instead).
We prove that these systems can satisfy properties of expansive systems or not.
For instance, unlike positively expansive maps [3],
positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces.
We also prove
that a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it is
so outside finitely many points.
Finally, we prove that a homeomorphism on a compact metric space
is $n$-expansive if and only if it is so outside finitely many orbits.
These last resuls extends previous ones for expansive systems [2],[11],[12].

A generalization of expansivity

/pdf/on-expansive-transformation-groups-bmu8fat95e.pdf

On expansive transformation groups

In this paper we study N-expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M with non-wandering set Ω(f) being the whole of M then f is expansive. This condition on the non-wandering 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.

http://premat.fing.edu.uy/papers/2013/164.pdf

N-expansive homeomorphisms on surfaces.

R. D. Anderson [ 1 ] has shown that a homeomorphism of a compact, separable, metric space onto itself can be lifted to a homeomorphism of a zero-dimensional space onto itself. He used a generalized shift transformation in his construction. This construction will not always lift expansive homeomorphisms (defined in [:3, 10.31])to expansive homeomorphisms, nor will the generalized shift reduce to a (subspace restriction of a) symbolic flow (see [3, 1.32 and 12.11 ]). It is the purpose of this note to prove that a homeomorphism of a compact metric space onto itself is expansive if and only if it can be lifted into a symbolic flow by a certain sort of map. Let X denote a compact metric (d) space and f a homeomorphism of X onto itself. Let (Sn, o-) denote the symbolic flow on n symbols [3, 12.15]. We say that (X,f) is lifted into (S,,, o') by p if there exists a subset W of Sn such that p is a continuous function from W onto X and p f = o'p (functions written on the right). I fA = {U1, • • • , Un} is a finite open cover of X, we define a function p from S~ into the collection of subsets of X as follows. Let a be a bisequence in S~ with a(i) denoting the ith entry. Then x is in ap if and only if xfis in cl Ua~o for each integer i. Here cl denotes the closure operator. The function p is called the characteristic correspondence f o r f a n d A. If each ap is empty or a singleton we consider the range of p to be X, and the domain to be {a: ap ~ o}.

Lifting expansive homeomorphisms to symbolic flows

Two-sided matching and game on investing in carbon emission reduction technology under a cap-and-trade system

(1962). Expansive Self-Homeomorphisms of a Compact Metric Space. The American Mathematical Monthly: Vol. 69, No. 5, pp. 386-391.

B. F. Bryant

Papers

Expansive Self-Homeomorphisms of a Compact Metric Space