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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク wandering set ： ウィキペディア英語版 wandering set In those branches of mathematics called dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing in such systems. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is very much the opposite of a conservative system, for which the ideas of the Poincaré recurrence theorem apply. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927. ==Wandering points== A common, discrete-time definition of wandering sets starts with a map $f:X\backslash to\; X$ of a topological space ''X''. A point $x\backslash in\; X$ is said to be a wandering point if there is a neighbourhood ''U'' of ''x'' and a positive integer ''N'' such that for all $n>N$, the iterated map is non-intersecting: :$f^n(U)\; \backslash cap\; U\; =\; \backslash varnothing.\backslash ,$ A handier definition requires only that the intersection have measure zero. To be precise, the definition requires that ''X'' be a measure space, i.e. part of a triple $(X,\backslash Sigma,\backslash mu)$ of Borel sets $\backslash Sigma$ and a measure $\backslash mu$ such that :$\backslash mu\backslash left(f^n(U)\; \backslash cap\; U\; \backslash right)\; =\; 0.\backslash ,$ Similarly, a continuous-time system will have a map $\backslash varphi\_t:X\backslash to\; X$ defining the time evolution or flow of the system, with the time-evolution operator $\backslash varphi$ being a one-parameter continuous abelian group action on ''X'': :$\backslash varphi\_\; =\; \backslash varphi\_t\; \backslash circ\; \backslash varphi\_s.\backslash ,$ In such a case, a wandering point $x\backslash in\; X$ will have a neighbourhood ''U'' of ''x'' and a time ''T'' such that for all times $t>T$, the time-evolved map is of measure zero: :$\backslash mu\backslash left(\backslash varphi\_t(U)\; \backslash cap\; U\; \backslash right)\; =\; 0.\backslash ,$ These simpler definitions may be fully generalized to the group action of a topological group. Let $\backslash Omega=(X,\backslash Sigma,\backslash mu)$ be a measure space, that is, a set with a measure defined on its Borel subsets. Let $\backslash Gamma$ be a group acting on that set. Given a point $x\; \backslash in\; \backslash Omega$, the set :$\backslash$ is called the trajectory or orbit of the point ''x''. An element $x\; \backslash in\; \backslash Omega$ is called a wandering point if there exists a neighborhood ''U'' of ''x'' and a neighborhood ''V'' of the identity in $\backslash Gamma$ such that :$\backslash mu\backslash left(\backslash gamma\; \backslash cdot\; U\; \backslash cap\; U\backslash right)=0$ for all $\backslash gamma\; \backslash in\; \backslash Gamma-V$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「wandering set」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.