Expansive homeomorphism

Definition

If (X,d) is a metric space, a homeomorphism f\colon X\to X is said to be expansive if there is a constant

:\varepsilon_00,

called the expansivity constant, such that for every pair of points x\neq y in X there is an integer n such that

:d(f^n(x),f^n(y))\geq\varepsilon_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

:f\colon X\to 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 x\neq y in X, there is an n\in\mathbb{N} such that d(f^n(x),f^n(y))\geq \varepsilon_0.

Theorem of uniform expansivity

Given f an expansive homeomorphism of a compact metric space, the theorem of uniform expansivity states that for every \epsilon0 and \delta0 there is an N0 such that for each pair x,y of points of X such that d(x,y)\epsilon, there is an n\in \mathbb{Z} with \vert n\vert\leq 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).