# Expansive

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 $\left(X,d\right)$ is a metric space, a homeomorphism $fcolon X o X$ is said to be expansive if there is a constant

:$varepsilon_0>0,$

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

:$d\left(f^n\left(x\right),f^n\left(y\right)\right)geqvarepsilon_0$.

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 thatassumption expansivity is a topological property; i.e. if $d\text{'}$ is any other metric generating the same topology as $d$, and if $f$ is expansive in $\left(X,d\right)$, then $f$ is expansive in $\left(X,d\text{'}\right)$ (possibly with a different expansivity constant).

If

:$fcolon X o X$

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

:$varepsilon_0$

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

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 $d\left(x,y\right)>epsilon$, there is an $nin mathbb\left\{Z\right\}$ with $vert nvertleq N$ such that

:$d\left(f^n\left(x\right),f^n\left(y\right)\right) > c-delta,$

where $c$ is the expansivity constant of $f$ ( [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4678 proof] ).

Discussion

Positive expansivity is much stronger than expansivity. In fact, one can prove that if $X$ is compact and $f$ is a positivelyexpansive homeomorphism, then $X$ is finite ( [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4677 proof] ).

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• Expansive — Ex*pan sive ([e^]ks*p[a^]n s[i^]v), a. [Cf. F. expansif.] 1. Having a capacity or tendency to expand or dilate; diffusive; of much expanse; wide extending; as, the expansive force of heat; the expansive quality of air. [WordNet sense 1] [1913… …   The Collaborative International Dictionary of English

• expansive — [adj1] broad, comprehensive all embracing, ample, big, dilatant, elastic, expanding, expansile, extensive, far reaching, great, inclusive, large, scopic, scopious, stretching, thorough, unrepressed, unsuppressed, voluminous, wide, wide ranging,… …   New thesaurus

• expansive — [ek span′siv] adj. [< L expansus (see EXPANSE) + IVE] 1. tending or being able to expand 2. of, or working by means of, expansion 3. widely extended; broad; extensive; comprehensive 4. characterized by a free and generous nature; sympathetic;… …   English World dictionary

• expansive — index ample, broad, capacious, complete (all embracing), comprehensive, extensive, omnibus, open ended …   Law dictionary

• expansive — 1650s, tending to expand, from L. expans (see EXPANSE (Cf. expanse)) + IVE (Cf. ive). Related: Expansively; expansiveness …   Etymology dictionary

• expansive — *elastic, resilient, buoyant, volatile, effervescent Analogous words: exuberant, luxuriant, lavish, prodigal (see PROFUSE): generous, *liberal, bountiful, bounteous, open handed: exalted, magnified, aggrandized (see EXALT) Antonyms: tense:… …   New Dictionary of Synonyms

• expansive — ► ADJECTIVE 1) covering a wide area; extensive. 2) relaxed, genial, and communicative. 3) tending towards territorial expansion. DERIVATIVES expansively adverb expansiveness noun …   English terms dictionary

• expansive — [[t]ɪkspæ̱nsɪv[/t]] 1) ADJ: ADJ n If something is expansive, it covers or includes a large area or many things. [FORMAL] There also are several swing sets and an expansive grassy play area... They have played an expansive style of rugby. 2) ADJ… …   English dictionary

• expansive — ex|pan|sive [ıkˈspænsıv] adj 1.) very friendly and willing to talk a lot ▪ Hauser was in an expansive mood . 2.) very large in area, or using a lot of space ▪ expansive beaches ▪ She flung her arms out in an expansive gesture. 3.) including a lot …   Dictionary of contemporary English

• expansive — adjective Date: 1651 1. having a capacity or a tendency to expand 2. causing or tending to cause expansion 3. a. characterized by high spirits, generosity, or readiness to talk ; open < grew expansive after dinner > b. marked by or indicative of… …   New Collegiate Dictionary