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Maßerhaltende Abbildungen, manchmal auch maßtreue Abbildungen genannt, sind Selbstabbildungen eines Maßraums, die das Maß erhalten. Man spricht auch von maßerhaltenden dynamischen Systemen, insbesondere wenn man das Verhalten der Abbildung unter Iteration betrachtet. Umgekehrt spricht man von einem invarianten Maß einer Abbildung oder eines dynamischen Systems, wenn die Abbildung (oder das dynamische System) das Maß erhält. Maßerhaltende Abbildungen sind das Thema der Ergodentheorie innerhalb der Theorie der dynamischen Systeme. (de) In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium. (en) Un système dynamique mesuré est un objet mathématique, représentant un espace de phases muni d'une loi d'évolution, particulièrement étudié en théorie ergodique. (fr) 数学における測度保存力学系（そくどほぞんりきがくけい、英: measure-preserving dynamical system）は、力学系の抽象的形成や、特にエルゴード理論に現れる一研究対象である。 (ja) Em matemática, um sistema dinâmico preservando medida é um objeto de estudo na abstrata formulação de sistemas dinâmicos e em particular na teoria ergódica. Sistemas dinâmicos que preservam uma medida obedecem o , e são um caso especial dos sistemas conservativos. Eles fornecem a base matemática formal para uma ampla gama de sistemas físicos e, em particular, muitos sistemas da mecânica clássica (em particular, a maioria dos sistemas não dissipativos), bem como sistemas em equilíbrio termodinâmico. (pt) 保测动力系统（英語：Measure-preserving dynamical system）由一个概率空间和其上的一个保测变换組成。也就是说，它是一个系统 带有下面结构： * X是一个集合， * 是X上的一个σ-代数， * 是一个概率测度， * 为一个可测变换, 并且保持测度，即对任意 满足. 此时，称T为概率空间上的一个保测变换。 (zh) |
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http://www.math.nyu.edu/~lsy/papers/entropy.pdf http://www.math.nyu.edu/~lsy/papers/entropy.ps https://arxiv.org/abs/nlin/0208048 |
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Maßerhaltende Abbildungen, manchmal auch maßtreue Abbildungen genannt, sind Selbstabbildungen eines Maßraums, die das Maß erhalten. Man spricht auch von maßerhaltenden dynamischen Systemen, insbesondere wenn man das Verhalten der Abbildung unter Iteration betrachtet. Umgekehrt spricht man von einem invarianten Maß einer Abbildung oder eines dynamischen Systems, wenn die Abbildung (oder das dynamische System) das Maß erhält. Maßerhaltende Abbildungen sind das Thema der Ergodentheorie innerhalb der Theorie der dynamischen Systeme. (de) In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium. (en) Un système dynamique mesuré est un objet mathématique, représentant un espace de phases muni d'une loi d'évolution, particulièrement étudié en théorie ergodique. (fr) 数学における測度保存力学系（そくどほぞんりきがくけい、英: measure-preserving dynamical system）は、力学系の抽象的形成や、特にエルゴード理論に現れる一研究対象である。 (ja) Em matemática, um sistema dinâmico preservando medida é um objeto de estudo na abstrata formulação de sistemas dinâmicos e em particular na teoria ergódica. Sistemas dinâmicos que preservam uma medida obedecem o , e são um caso especial dos sistemas conservativos. Eles fornecem a base matemática formal para uma ampla gama de sistemas físicos e, em particular, muitos sistemas da mecânica clássica (em particular, a maioria dos sistemas não dissipativos), bem como sistemas em equilíbrio termodinâmico. (pt) 保测动力系统（英語：Measure-preserving dynamical system）由一个概率空间和其上的一个保测变换組成。也就是说，它是一个系统 带有下面结构： * X是一个集合， * 是X上的一个σ-代数， * 是一个概率测度， * 为一个可测变换, 并且保持测度，即对任意 满足. 此时，称T为概率空间上的一个保测变换。 (zh) |
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Maßerhaltende Abbildung (de) Système dynamique mesuré (fr) Measure-preserving dynamical system (en) 測度保存力学系 (ja) Sistema dinâmico preservando medida (pt) 保测动力系统 (zh) |
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