Expansive

Expansive

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In mathematics, the notion of expansivity formalizes the notion of points moving away from one-another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definition of positive expansivity, below, as well as the Schwarz-Ahlfors-Pick theorem demonstrate.

Definition

If (X,d) is a metric space, a homeomorphism fcolon Xto X is said to be expansive if there is a constant

varepsilon_0>0,

called the expansivity constant, such that for any pair of points xneq y in X there is an integer n such that

d(f^n(x),f^n(y))geqvarepsilon_0.

Note that in this definition, n can be positive or negative, and so f may be expansive in the forward or backward directions.

The space X is often assumed to be compact, since under that assumption expansivity is a topological property; i.e. if d' is any other metric generating the same topology as d, and if f is expansive in (X,d), then f is expansive in (X,d') (possibly with a different expansivity constant).

If

fcolon Xto X

is a continuous map, we say that X is positively expansive (or forward expansive) if there is a

varepsilon_0

such that, for any xneq y in X, there is an ninmathbb{N} such that d(f^n(x),f^n(y))geq varepsilon_0.

Theorem of uniform expansivity

Given f an expansive homeomorphism, the theorem of uniform expansivity states that for every epsilon>0 and delta>0 there is an N>0 such that for each pair x,y of points of X such that d(x,y)>epsilon, there is an nin mathbb{Z} with vert nvertleq N such that

d(f^n(x),f^n(y)) > c-delta,

where c is the expansivity constant of f (proof).

Discussion

Positive expansivity is much stronger than expansivity. In fact, one can prove that if X is compact and f is a positively expansive homeomorphism, then X is finite (proof).

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