# Dictionary Definition

expansive adj

1 able or tending to expand or characterized by
expansion; "Expansive materials"; "the expansive force of fire"
[ant: unexpansive]

2 impressive in scale; "an expansive lifestyle";
"in the grand manner" [syn: grand]

3 marked by exaggerated feelings of euphoria and
delusions of grandeur

4 friendly and open and willing to talk; "wine
made the guest expansive" [syn: talkative]

# User Contributed Dictionary

## English

# Extensive Definition

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- \varepsilon_0>0,

called the expansivity constant, such that for
any two points of X, their n-th
iterates are at least \varepsilon_0 apart for some integer n;
i.e. if for any pair of points x\neq y in X there is n\in\mathbb
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 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 n\in \mathbb 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).# Synonyms, Antonyms and Related Words

accessible, affable, airy, all jaw, all-embracing,
all-inclusive, amiable,
ample, amplitudinous, approachable, big, bouncy, broad, buoyant, candid, capacious, chatty, commodious, communicable, communicative, comprehensive, conversable, conversational, copious, deep, demonstrative, dilatable, dilatant, distensive, easy, effervescent, effusive, elastic, enlarging, expandable, expanding, expansible, expansile, expansional, extendable, extended, extending, extensible, extensile, extensional, extensive, extroverted, far-reaching,
flip, fluent, frank, free, free-speaking, free-spoken,
free-tongued, friendly,
full, gabby, garrulous, gassy, generous, genial, glib, gossipy, great, gregarious, gushy, infinite, inflatable, inflationary, large, lavish, liberal, long-winded, loquacious, multiloquent, multiloquious, newsy, open, openhanded, outgoing, outspoken, overtalkative, prolix, resilient, roomy, scopic, self-revealing,
self-revelatory, smooth,
sociable, spacious, spreading, talkative, talky, unconstrained, unhampered, unrepressed, unreserved, unrestrained, unrestricted, unreticent, unsecretive, unshrinking, unsilent, unsuppressed, vast, verbose, volatile, voluble, voluminous, warm, wide, wide-ranging, widespread, windy