Expansive 5508726 91763424 2006-12-03T10:07:14Z Rjwilmsi 203434 [[User:Wmahan/Spelling|sp]]: anyother->any other 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 <math>(X,d)</math> is a [[metric space]], a [[homeomorphism]] <math>f\colon X\to X</math> is said to be '''expansive''' if there is a constant :<math>\varepsilon_0>0,</math> called the '''expansivity constant''', such that for any two points of <math>X</math>, their [[iterated function|''n''-th iterates]] are at least <math>\varepsilon_0</math> apart for some integer <math>n</math>; i.e. if for any pair of points <math>x\neq y</math> in <math>X</math> there is <math>n\in\mathbb{Z}</math> such that :<math>d(f^n(x),f^n(y))\geq\varepsilon_0</math>. Note that in this definition, <math>n</math> can be positive or negative, and so <math>f</math> may be expansive in the forward or backward directions. The space <math>X</math> is often assumed to be [[compact]], since under that assumption expansivity is a topological property; i.e. if <math>d'</math> is any other metric generating the same topology as <math>d</math>, and if <math>f</math> is expansive in <math>(X,d)</math>, then <math>f</math> is expansive in <math>(X,d')</math> (possibly with a different expansivity constant). If :<math>f\colon X\to X</math> is a continuous map, we say that <math>X</math> is '''positively expansive''' (or '''forward expansive''') if there is a :<math>\varepsilon_0</math> such that, for any <math>x\neq y</math> in <math>X</math>, there is an <math>n\in\mathbb{N}</math> such that <math>d(f^n(x),f^n(y))\geq \varepsilon_0</math>. ==Theorem of uniform expansivity== Given ''f'' an expansive homeomorphism, the theorem of uniform expansivity states that for every <math>\epsilon>0</math> and <math>\delta>0</math> there is an <math>N>0</math> such that for each pair <math>x,y</math> of points of <math>X</math> such that <math>d(x,y)>\epsilon</math>, there is an <math>n\in \mathbb{Z}</math> with <math>\vert n\vert\leq N</math> such that :<math>d(f^n(x),f^n(y)) > c-\delta,</math> where <math>c</math> is the expansivity constant of <math>f</math> ([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 <math>X</math> is compact and <math>f</math> is a positively expansive homeomorphism, then <math>X</math> is finite ([http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4677 proof]). {{planetmath|id=4513|title=expansive}} {{planetmath|id=4678|title=uniform expansivity}} [[Category:Dynamical systems]]