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.


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


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


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' 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).


:fcolon X o X

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


such that, for any x eq 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 ( [http://planetmath.org/?op=getobj&from=objects&id=4678 proof] ).


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&from=objects&id=4677 proof] ).

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  • 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

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