uniform expansivity

Let (X,d) be a compactPlanetmathPlanetmath metric space and let f:XX be an expansive homeomorphismPlanetmathPlanetmath.

Theorem (uniform expansivity). For every ϵ>0 and δ>0 there is N>0 such that for each pair x,y of points of X such that d(x,y)>ϵ there is n with |n|N such that d(fn(x),fn(y))>c-δ, where c is the expansivity constant of f.

Proof. Let K={(x,y)X×X:d(x,y)ϵ/2}. Then K is closed, and hence compact. For each pair (x,y)K, there is n(x,y) such that d(fn(x,y)(x),fn(x,y)(y))c. Since the mapping F:X×XX×X defined by F(x,y)=(f(x),f(y)) is continuous, Fnx is also continuous and there is a neighborhoodMathworldPlanetmathPlanetmath U(x,y) of each (x,y)K such that d(fn(x,y)(u),fn(x,y)(v))<c-δ for each (u,v)U(x,y). Since K is compact and {U(x,y):(x,y)K} is an open cover of K, there is a finite subcover {U(xi,yi):1im}. Let N=max{|n(xi,yi)|:1im}. If d(x,y)>ϵ, then (x,y)K, so that (x,y)U(xi,yi) for some i{1,,m}. Thus for n=n(xi,yi) we have d(fn(x),fn(y))<c-δ and |n|N as requred.

Title uniform expansivity
Canonical name UniformExpansivity
Date of creation 2013-03-22 13:55:15
Last modified on 2013-03-22 13:55:15
Owner Koro (127)
Last modified by Koro (127)
Numerical id 7
Author Koro (127)
Entry type Theorem
Classification msc 37B99