Expansive

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 $f\colon X\to X$ is said to be expansive if there is a constant

ε₀ > 0,

called the expansivity constant, such that for any 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

ε₀

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, the theorem of uniform expansivity states that for every ε > 0 and δ > 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 $n\in \mathbb{Z}$ with $\vert n\vert\leq N$ such that

d(fⁿ(x),fⁿ(y)) > c − δ,

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).

External links

• Expansive dynamical systems on scholarpedia

This article incorporates material from expansive on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from uniform expansivity on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.