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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.

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.

在数学中，保测动力系统是动力系统，尤其是遍历理论的抽象表述中的研究对象。保测度系统服从庞加莱始态复现定理，是保守系统的一个特例。它们为广泛的物理系统提供了形式化的数学基础，特别是许多经典力学系统(特别是大多数非耗散系统)以及热力学平衡系统。

Definition

A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system

= = 定义 = = 一个保测动力系统被定义为一个概率空间和一个保测度变换。更详细地说，它是一个系统

[math]\displaystyle{ (X, \mathcal{B}, \mu, T) }[/math]

(X, \mathcal{B}, \mu, T)

(x，数学{ b } ，mu，t)

with the following structure:

有以下结构:

[math]\displaystyle{ X }[/math] is a set,
[math]\displaystyle{ \mathcal B }[/math] is a σ-algebra over [math]\displaystyle{ X }[/math],
[math]\displaystyle{ \mu:\mathcal{B}\rightarrow[0,1] }[/math] is a probability measure, so that [math]\displaystyle{ \mu (X) = 1 }[/math], and [math]\displaystyle{ \mu(\varnothing) = 0 }[/math],
[math]\displaystyle{ T:X \rightarrow X }[/math] is a measurable transformation which preserves the measure [math]\displaystyle{ \mu }[/math], i.e., [math]\displaystyle{ \forall A\in \mathcal{B}\;\; \mu(T^{-1}(A))=\mu(A) }[/math].

X is a set,
\mathcal B is a σ-algebra over X,
\mu:\mathcal{B}\rightarrow[0,1] is a probability measure, so that \mu (X) = 1, and \mu(\varnothing) = 0,
T:X \rightarrow X is a measurable transformation which preserves the measure \mu, i.e., \forall A\in \mathcal{B}\;\; \mu(T^{-1}(A))=\mu(A) .

x 是一个集合，
mathcal b 是 x 上的 σ- 代数，
mu: mathcal { b } right tarrow [0,1]是一个机率量测，因此 mu (x) = 1，mu (varnothing) = 0，
t: x right tarrow x 是一个可测变换，它保留了测度 μ，即数学{ b }中的所有 a; mu (t ^ {-1}(a) = mu (a)。

Discussion

One may ask why the measure preserving transformation is defined in terms of the inverse [math]\displaystyle{ \mu(T^{-1}(A))=\mu(A) }[/math] instead of the forward transformation [math]\displaystyle{ \mu(T(A))=\mu(A) }[/math]. This can be understood in a fairly easy fashion. Consider a mapping [math]\displaystyle{ \mathcal{T} }[/math] of power sets:

[math]\displaystyle{ \mathcal{T}:P(X)\to P(X) }[/math]

Consider now the special case of maps [math]\displaystyle{ \mathcal{T} }[/math] which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends [math]\displaystyle{ X }[/math] to [math]\displaystyle{ X }[/math] (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map [math]\displaystyle{ T:X\to X }[/math] by writing [math]\displaystyle{ \mathcal{T}(A)=T^{-1}(A) }[/math]. Of course, one could also define [math]\displaystyle{ \mathcal{T}(A)=T(A) }[/math], but this is not enough to specify all such possible maps [math]\displaystyle{ \mathcal{T} }[/math]. That is, conservative, Borel-preserving maps [math]\displaystyle{ \mathcal{T} }[/math] cannot, in general, be written in the form [math]\displaystyle{ \mathcal{T}(A)=T(A). }[/math] Obviously! one might say; consider, for example, the map of the unit interval [math]\displaystyle{ T:[0,1) \to [0,1) }[/math] given by [math]\displaystyle{ x \mapsto 2x\mod 1; }[/math] this is the Bernoulli map.

One may ask why the measure preserving transformation is defined in terms of the inverse \mu(T^{-1}(A))=\mu(A) instead of the forward transformation \mu(T(A))=\mu(A). This can be understood in a fairly easy fashion. Consider a mapping \mathcal{T} of power sets:

\mathcal{T}:P(X)\to P(X)

Consider now the special case of maps \mathcal{T} which preserve intersections, unions and complements (so that it is a map of Borel sets) and also sends X to X (because we want it to be conservative). Every such conservative, Borel-preserving map can be specified by some surjective map T:X\to X by writing \mathcal{T}(A)=T^{-1}(A). Of course, one could also define \mathcal{T}(A)=T(A), but this is not enough to specify all such possible maps \mathcal{T}. That is, conservative, Borel-preserving maps \mathcal{T} cannot, in general, be written in the form \mathcal{T}(A)=T(A). Obviously! one might say; consider, for example, the map of the unit interval T:[0,1) \to [0,1) given by x \mapsto 2x\mod 1; this is the Bernoulli map.

= = 讨论 = = 有人可能会问，为什么保测度变换定义为倒数 mu (t ^ {-1}(a)) = mu (a) ，而不是正向变换 mu (t (a) = mu (a)。这一点可以很容易地理解。考虑幂集的映射数学{ t } : mathcal { t } : p (x) to p (x)现在考虑映射数学{ t }的特殊情况，它保留了交集、联合集和补集(所以它是 Borel 集的映射) ，并且将 x 发送给 x (因为我们希望它是保守的)。每一个这样的保守的 borel 保持映射都可以通过一些满射映射 t: x 到 x 来指定，方法是写数学函数{ t }(a) = t ^ {-1}(a)。当然，也可以定义数学函数{ t }(a) = t (a) ，但这不足以指定所有这些可能的映射数学函数{ t }。也就是说，保守的、保持 borel 的映射数学{ t }一般不能以数学{ t }(a) = t (a)的形式写出。很明显！例如，考虑单位间隔 t: [0,1]到[0,1]的映射，由 x 映射到2x 模1; 这是伯努利映射。

Note that [math]\displaystyle{ \mu(T^{-1}(A)) }[/math] has the form of a pushforward, whereas [math]\displaystyle{ \mu(T(A)) }[/math] is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map [math]\displaystyle{ T }[/math]; the measure [math]\displaystyle{ \mu }[/math] can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)

Note that \mu(T^{-1}(A)) has the form of a pushforward, whereas \mu(T(A)) is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in terms of the pushforward. For example, the transfer operator is defined in terms of the pushforward of the transformation map T; the measure \mu can now be understood as an invariant measure; it is just the Frobenius–Perron eigenvector of the transfer operator (recall, the FP eigenvector is the largest eigenvector of a matrix; in this case it is the eigenvector which has the eigenvalue one: the invariant measure.)

注意，mu (t ^ {-1}(a))具有前推的形式，而 mu (t (a))通常称为拉回。几乎所有的动力系统的性质和行为都是用前推来定义的。例如，转移算子是根据变换映射 t 的前推定义的; 测度 mu 现在可以理解为不变测度; 它只是转移算子的 Frobenius-Perron 特征向量(回想一下，FP 特征向量是矩阵的最大特征向量; 在这种情况下，它是具有特征值的特征向量: 不变测度。)

There are two classification problems of interest. One, discussed below, fixes [math]\displaystyle{ (X, \mathcal{B}, \mu) }[/math] and asks about the isomorphism classes of a transformation map [math]\displaystyle{ T }[/math]. The other, discussed in transfer operator, fixes [math]\displaystyle{ (X, \mathcal{B}) }[/math] and [math]\displaystyle{ T }[/math], and asks about maps [math]\displaystyle{ \mu }[/math] that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into dissipative systems and the route to equilibrium.

There are two classification problems of interest. One, discussed below, fixes (X, \mathcal{B}, \mu) and asks about the isomorphism classes of a transformation map T. The other, discussed in transfer operator, fixes (X, \mathcal{B}) and T, and asks about maps \mu that are measure-like. Measure-like, in that they preserve the Borel properties, but are no longer invariant; they are in general dissipative and so give insights into dissipative systems and the route to equilibrium.

有两个兴趣的分类问题。下面讨论的一个问题是，确定(x，数学{ b } ，mu) ，并询问转换映射 t 的同构类。另一个在传输运算符、修复程序(x，mathcal { b })和 t 中讨论，并询问关于类似度量值的映射 mu。类似测量，因为它们保持了 Borel 的性质，但不再是不变的; 它们一般是耗散的，因此可以洞察耗散系统和达到平衡的路径。

In terms of physics, the measure-preserving dynamical system [math]\displaystyle{ (X, \mathcal{B}, \mu, T) }[/math] often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map [math]\displaystyle{ T }[/math] describes this stirring, mixing, etc. then the system [math]\displaystyle{ (X, \mathcal{B}, \mu, T) }[/math] is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure [math]\displaystyle{ \mu }[/math] is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.

In terms of physics, the measure-preserving dynamical system (X, \mathcal{B}, \mu, T) often describes a physical system that is in equilibrium, for example, thermodynamic equilibrium. One might ask: how did it get that way? Often, the answer is by stirring, mixing, turbulence, thermalization or other such processes. If a transformation map T describes this stirring, mixing, etc. then the system (X, \mathcal{B}, \mu, T) is all that is left, after all of the transient modes have decayed away. The transient modes are precisely those eigenvectors of the transfer operator that have eigenvalue less than one; the invariant measure \mu is the one mode that does not decay away. The rate of decay of the transient modes are given by (the logarithm of) their eigenvalues; the eigenvalue one corresponds to infinite half-life.

在物理学方面，保测动力系统(x，数学{ b } ，mu，t)通常描述一个处于平衡状态的物理系统，例如，热力学平衡。有人可能会问: 怎么会变成这样？通常，答案是通过搅拌，混合，湍流，热化或其他类似的过程。如果一个变换图 t 描述了这种搅拌，混合等。在所有的瞬态模式衰减之后，系统(x，数学{ b } ，mu，t)就是剩下的全部。瞬态模式正是那些本征值小于1的转移算子的本征向量; 不变测度 μ 是不衰减的模式。瞬态模态的衰减率是由它们的特征值(对数)给出的; 特征值的衰减率对应于无限的半衰期。

Informal example

The microcanonical ensemble from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height [math]\displaystyle{ w\times l\times h, }[/math] consisting of [math]\displaystyle{ N }[/math] atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in [math]\displaystyle{ w\times l\times h\times \mathbb{R}^3. }[/math] A given collection of [math]\displaystyle{ N }[/math] atoms would then be a single point somewhere in the space [math]\displaystyle{ (w\times l\times h)^N \times \mathbb{R}^{3N}. }[/math] The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space [math]\displaystyle{ X }[/math] above.

The microcanonical ensemble from physics provides an informal example. Consider, for example, a fluid, gas or plasma in a box of width, length and height w\times l\times h, consisting of N atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in w\times l\times h\times \mathbb{R}^3. A given collection of N atoms would then be a single point somewhere in the space (w\times l\times h)^N \times \mathbb{R}^{3N}. The "ensemble" is the collection of all such points, that is, the collection of all such possible boxes (of which there are an uncountably-infinite number). This ensemble of all-possible-boxes is the space X above.

= = 非正式的例子 = = 来自物理学的微正则系综提供了一个非正式的例子。例如，考虑一个流体，气体或等离子体在一个盒子的宽度，长度和高度 w 乘以 l 乘以 h，由 n 个原子组成。那个盒子里的单个原子可能在任何地方，具有任意的速度; 它可以用 w 乘以 l 乘以 h 乘以 mathbb { r } ^ 3的单点来表示。一个给定的 n 原子集合就是空间中的某个点(w 乘以 l 乘以 h) ^ n 乘以 mathbb { r } ^ {3N }。“ ensemble”是所有这些点的集合，也就是所有这些可能的盒子的集合(其中有一个不可数的无限数)。这个所有可能的盒子的集合就是上面的空间 x。

In the case of an ideal gas, the measure [math]\displaystyle{ \mu }[/math] is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if [math]\displaystyle{ p_i(x,y,z,v_x,v_y,v_z)\,d^3x\,d^3p }[/math] is the probability of atom [math]\displaystyle{ i }[/math] having position and velocity [math]\displaystyle{ x,y,z,v_x,v_y,v_z }[/math], then, for [math]\displaystyle{ N }[/math] atoms, the probability is the product of [math]\displaystyle{ N }[/math] of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the Maxwell–Boltzmann measure. It will be enormously tiny, of order [math]\displaystyle{ \mathcal{O}\left(2^{-3N}\right). }[/math] Of all possible boxes in the ensemble, this is a ridiculously small fraction.

In the case of an ideal gas, the measure \mu is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if p_i(x,y,z,v_x,v_y,v_z)\,d^3x\,d^3p is the probability of atom i having position and velocity x,y,z,v_x,v_y,v_z, then, for N atoms, the probability is the product of N of these. This measure is understood to apply to the ensemble. So, for example, one of the possible boxes in the ensemble has all of the atoms on one side of the box. One can compute the likelihood of this, in the Maxwell–Boltzmann measure. It will be enormously tiny, of order \mathcal{O}\left(2^{-3N}\right). Of all possible boxes in the ensemble, this is a ridiculously small fraction.

在理想气体的情况下，测量 μ 由麦克斯韦-波兹曼分布给出。它是乘积测度，如果 p _ i (x，y，z，v _ x，v _ y，v _ z) ，d ^ 3x，d ^ 3p 是原子 i 具有位置和速度 x，y，z，v _ x，v _ y，v _ z 的概率，那么，对 n 个原子，概率就是它们的 n 的乘积。这一措施被理解为适用于整体。例如，集合中的一个可能的盒子中，所有的原子都在盒子的一边。我们可以用麦克斯韦-玻尔兹曼测量来计算这种可能性。它将是非常小的，有序的数学{ o }左(2 ^ {-3N }右)。在所有可能的盒子中，这是一个可笑的小部分。

The only reason that this is an "informal example" is because writing down the transition function [math]\displaystyle{ T }[/math] is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if the interaction is not an ideal-gas billiard-ball type interaction, but is instead a van der Waals interaction, or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the Maxwell–Boltzmann distribution. The art of physics is finding reasonable approximations.

The only reason that this is an "informal example" is because writing down the transition function T is difficult, and, even if written down, it is hard to perform practical computations with it. Difficulties are compounded if the interaction is not an ideal-gas billiard-ball type interaction, but is instead a van der Waals interaction, or some other interaction suitable for a liquid or a plasma; in such cases, the invariant measure is no longer the Maxwell–Boltzmann distribution. The art of physics is finding reasonable approximations.

这是一个“非正式示例”的唯一原因是，写下转换函数 t 很困难，而且即使写下来，也很难用它执行实际计算。如果这种相互作用不是理想气体台球-球类型的相互作用，而是范德瓦尔斯相互作用，或者其他适合于液体或等离子体的相互作用，那么这种相互作用就更加困难; 在这种情况下，不变测度就不再是麦克斯韦-波兹曼分布。物理学的艺术在于找到合理的近似值。

This system does exhibit one key idea from the classification of measure-preserving dynamical systems: two ensembles, having different temperatures, are inequivalent. The entropy for a given canonical ensemble depends on its temperature; as physical systems, it is "obvious" that when the temperatures differ, so do the systems. This holds in general: systems with different entropy are not isomorphic.

这个系统展示了保测度动力系统分类的一个关键思想: 两个不同温度的系综是不等价的。给定正则系综的熵取决于它的温度; 作为物理系统，很明显，当温度不同时，系统也不同。这在一般情况下是成立的: 具有不同熵的系统是不同构的。

Examples

文件:Exampleergodicmap.svg

Example of a (Lebesgue measure) preserving map: T : [0,1) → [0,1), [math]\displaystyle{ x \mapsto 2x \mod 1. }[/math]

Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed.

文件:Exampleergodicmap.svg

Example of a (Lebesgue measure) preserving map: T : [0,1) → [0,1), x \mapsto 2x \mod 1.

Unlike the informal example above, the examples below are sufficiently well-defined and tractable that explicit, formal computations can be performed.

= = 实例 = = thumb | 示例保存勒贝格测度: t: [0,1]→[0,1] ，x mapsto 2x mod 1. 与上面的非正式示例不同，下面的示例定义充分且易于处理，可以执行显式的正式计算。

μ could be the normalized angle measure dθ/2π on the unit circle, and T a rotation. See equidistribution theorem;
the Bernoulli scheme;
the interval exchange transformation;
with the definition of an appropriate measure, a subshift of finite type;
the base flow of a random dynamical system;
the flow of a Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the symplectic volume form) by Liouville's theorem (Hamiltonian);^[1]
for certain maps and Markov processes, the Krylov–Bogolyubov theorem establishes the existence of a suitable measure to form a measure-preserving dynamical system.

μ could be the normalized angle measure dθ/2π on the unit circle, and T a rotation. See equidistribution theorem;
the Bernoulli scheme;
the interval exchange transformation;
with the definition of an appropriate measure, a subshift of finite type;
the base flow of a random dynamical system;
the flow of a Hamiltonian vector field on the tangent bundle of a closed connected smooth manifold is measure-preserving (using the measure induced on the Borel sets by the symplectic volume form) by Liouville's theorem (Hamiltonian);
for certain maps and Markov processes, the Krylov–Bogolyubov theorem establishes the existence of a suitable measure to form a measure-preserving dynamical system.

μ 可以是单位圆上的归一化角度 dθ/2π，t 为旋转角度。参见等分布定理;
Bernoulli 格式;
区间交换变换;
定义了适当的测度，有限类型的子移位;
随机动力系统的基本流;
闭连通光滑流形的切丛上的哈密顿向量场流是保测度的(利用辛体积形式在 Borel 集上诱导的测度) ，由 Liouville 定理(Hamiltonian) ;
对于某些地图和 Markov 过程，Krylov-Bogolyubov 定理建立了形成保测动力系统的适当测度的存在性。

Generalization to groups and monoids

The definition of a measure-preserving dynamical system can be generalized to the case in which T is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group, in which case we have the action of a group upon the given probability space) of transformations T_s : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation T_s satisfies the same requirements as T above.^[1] In particular, the transformations obey the rules:

[math]\displaystyle{ T_0 = \mathrm{id}_X :X \rightarrow X }[/math], the identity function on X;
[math]\displaystyle{ T_{s} \circ T_{t} = T_{t + s} }[/math], whenever all the terms are well-defined;
[math]\displaystyle{ T_{s}^{-1} = T_{-s} }[/math], whenever all the terms are well-defined.

The definition of a measure-preserving dynamical system can be generalized to the case in which T is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group, in which case we have the action of a group upon the given probability space) of transformations Ts : X → X parametrized by s ∈ Z (or R, or N ∪ {0}, or [0, +∞)), where each transformation Ts satisfies the same requirements as T above. In particular, the transformations obey the rules:

T_0 = \mathrm{id}_X :X \rightarrow X, the identity function on X;
T_{s} \circ T_{t} = T_{t + s}, whenever all the terms are well-defined;
T_{s}^{-1} = T_{-s}, whenever all the terms are well-defined.

= = 对群和幺半群的推广 = = 一个保测动力系统的定义可以推广到这样的情形: t 不是一个单一的迭代给出系统动力学的变换，而是一个幺半群(或者甚至是一个群，在这种情形下我们有一个群对给定概率空间的作用)的变换 Ts: x → x 被 s ∈ z (或 r，或 n ∪{0} ，或者[0，+ ∞)参数化，其中每个变换 Ts 满足上面 t 的相同要求。特别地，这些变换遵循以下规则:

t _ 0 = mathrm { id } _ x: x right tarrow x，x 上的恒等函数;
t _ { s } circ t _ { t } = t _ { t + s } ，只要所有的术语都定义良好;
t _ { s } ^ {-1} = t _ {-s } ，只要所有的术语都定义良好。

The earlier, simpler case fits into this framework by defining T_s = T^s for s ∈ N.

The earlier, simpler case fits into this framework by defining Ts = Ts for s ∈ N.

更早的，更简单的情形通过定义 s ∈ n 的 Ts = Ts 符合这个框架。

Homomorphisms

The concept of a homomorphism and an isomorphism may be defined.

= = = 同态 = = 同态和同构的概念可以定义。

Consider two dynamical systems [math]\displaystyle{ (X, \mathcal{A}, \mu, T) }[/math] and [math]\displaystyle{ (Y, \mathcal{B}, \nu, S) }[/math]. Then a mapping

Consider two dynamical systems (X, \mathcal{A}, \mu, T) and (Y, \mathcal{B}, \nu, S). Then a mapping

考虑两个动力系统(x，数学{ a } ，mu，t)和(y，数学{ b } ，nu，s)。然后是一个映射

[math]\displaystyle{ \varphi:X \to Y }[/math]

\varphi:X \to Y

从 x 到 y

is a homomorphism of dynamical systems if it satisfies the following three properties:

is a homomorphism of dynamical systems if it satisfies the following three properties:

是动力系统的同态，如果它满足以下三个性质:

The map [math]\displaystyle{ \varphi\ }[/math] is measurable.
For each [math]\displaystyle{ B \in \mathcal{B} }[/math], one has [math]\displaystyle{ \mu (\varphi^{-1}B) = \nu(B) }[/math].
For [math]\displaystyle{ \mu }[/math]-almost all [math]\displaystyle{ x \in X }[/math], one has [math]\displaystyle{ \varphi(Tx) = S(\varphi x) }[/math].

The map \varphi\ is measurable.
For each B \in \mathcal{B}, one has \mu (\varphi^{-1}B) = \nu(B).
For \mu-almost all x \in X, one has \varphi(Tx) = S(\varphi x).

这张地图是可以测量的。# 对于数学{ b }中的每个 b，有一个 μ (varphi ^ {-1} b) = nu (b)。# 对于 mu-几乎所有 x 在 x 中，一个有 varphi (Tx) = s (varphi x)。

The system [math]\displaystyle{ (Y, \mathcal{B}, \nu, S) }[/math] is then called a factor of [math]\displaystyle{ (X, \mathcal{A}, \mu, T) }[/math].

The system (Y, \mathcal{B}, \nu, S) is then called a factor of (X, \mathcal{A}, \mu, T).

这个系统(y，mathcal { b } ，nu，s)然后被称为(x，mathcal { a } ，mu，t)的因子。

The map [math]\displaystyle{ \varphi\; }[/math] is an isomorphism of dynamical systems if, in addition, there exists another mapping

The map \varphi\; is an isomorphism of dynamical systems if, in addition, there exists another mapping

映射 varphi 是动力系统的同构，如果存在另一个映射的话

[math]\displaystyle{ \psi:Y \to X }[/math]

\psi:Y \to X

psi: y to x

that is also a homomorphism, which satisfies

这也是一个同态

for [math]\displaystyle{ \mu }[/math]-almost all [math]\displaystyle{ x \in X }[/math], one has [math]\displaystyle{ x = \psi(\varphi x) }[/math];
for [math]\displaystyle{ \nu }[/math]-almost all [math]\displaystyle{ y \in Y }[/math], one has [math]\displaystyle{ y = \varphi(\psi y) }[/math].

for \mu-almost all x \in X, one has x = \psi(\varphi x);
for \nu-almost all y \in Y, one has y = \varphi(\psi y).

对 mu-几乎所有 x 在 x 中，一个是 x = psi (varphi x) ; 对 nu-几乎所有 y 在 y 中，一个是 y = varphi (psi y)。

Hence, one may form a category of dynamical systems and their homomorphisms.

因此，可以形成一类动力系统及其同态。

Generic points

A point x ∈ X is called a generic point if the orbit of the point is distributed uniformly according to the measure.

A point x ∈ X is called a generic point if the orbit of the point is distributed uniformly according to the measure.

= = = 一般点 = = 一点 x ∈ x 称为一般点，如果该点的轨道按照测度均匀分布。

Symbolic names and generators

Consider a dynamical system [math]\displaystyle{ (X, \mathcal{B}, T, \mu) }[/math], and let Q = {Q₁, ..., Q_k} be a partition of X into k measurable pair-wise disjoint pieces. Given a point x ∈ X, clearly x belongs to only one of the Q_i. Similarly, the iterated point Tⁿx can belong to only one of the parts as well. The symbolic name of x, with regards to the partition Q, is the sequence of integers {a_n} such that

Consider a dynamical system (X, \mathcal{B}, T, \mu), and let Q = {Q1, ..., Qk} be a partition of X into k measurable pair-wise disjoint pieces. Given a point x ∈ X, clearly x belongs to only one of the Qi. Similarly, the iterated point Tnx can belong to only one of the parts as well. The symbolic name of x, with regards to the partition Q, is the sequence of integers {an} such that

= = = 符号名和生成元 = = 考虑一个动力系统(x，数学{ b } ，t，mu) ，设 q = { Q1，... ，Qk }是 x 的一个分区，成为 k 个可测的成对不交片。给定一个点 x ∈ x，显然 x 只属于其中的一个。类似地，迭代点 Tnx 也只能属于其中一个部分。对于分区 q，x 的符号名称是整数{ an }的序列，这样的话

[math]\displaystyle{ T^nx \in Q_{a_n}. }[/math]

T^nx \in Q_{a_n}.

t ^ nx in q _ { a _ n }.

The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. A partition Q is called a generator or generating partition if μ-almost every point x has a unique symbolic name.

The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. A partition Q is called a generator or generating partition if μ-almost every point x has a unique symbolic name.

对于一个分区而言，这套符号名称称为符号动力学动力系统。如果 μ 几乎每个点 x 都有唯一的符号名，则分区 q 称为生成器或生成分区。

Operations on partitions

Given a partition Q = {Q₁, ..., Q_k} and a dynamical system [math]\displaystyle{ (X, \mathcal{B}, T, \mu) }[/math], define the T-pullback of Q as

Given a partition Q = {Q1, ..., Qk} and a dynamical system (X, \mathcal{B}, T, \mu), define the T-pullback of Q as

= = = 分区上的操作 = = 给定一个分区 q = { Q1，... ，Qk }和一个动力系统(x，数学{ b } ，t，mu) ，定义 q 的 T-pullback 为

[math]\displaystyle{ T^{-1}Q = \{T^{-1}Q_1,\ldots,T^{-1}Q_k\}. }[/math]

T^{-1}Q = \{T^{-1}Q_1,\ldots,T^{-1}Q_k\}.

t ^ {-1} q = { t ^ {-1} q1，ldots，t ^ {-1} qk }.

Further, given two partitions Q = {Q₁, ..., Q_k} and R = {R₁, ..., R_m}, define their refinement as

Further, given two partitions Q = {Q1, ..., Qk} and R = {R1, ..., Rm}, define their refinement as

进一步，给定两个分区 q = { Q1，... ，Qk }和 r = { R1，... ，Rm } ，将它们的求精定义为

[math]\displaystyle{ Q \vee R = \{Q_i \cap R_j \mid i=1,\ldots,k,\ j=1,\ldots,m,\ \mu(Q_i \cap R_j) \gt 0 \}. }[/math]

Q \vee R = \{Q_i \cap R_j \mid i=1,\ldots,k,\ j=1,\ldots,m,\ \mu(Q_i \cap R_j) > 0 \}.

qvee r = { q_i cap r_j mid i = 1，ldots，k，j = 1，ldots，m，mu (q_i cap r_j) > 0}.

With these two constructs, the refinement of an iterated pullback is defined as

With these two constructs, the refinement of an iterated pullback is defined as

使用这两个构造，迭代 pullback 的精化定义为

[math]\displaystyle{ \begin{align} \bigvee_{n=0}^N T^{-n}Q & = \{Q_{i_0} \cap T^{-1}Q_{i_1} \cap \cdots \cap T^{-N}Q_{i_N} \\ & {} \qquad \mbox { where }i_\ell = 1,\ldots,k ,\ \ell=0,\ldots,N,\ \\ & {} \qquad \qquad \mu \left (Q_{i_0} \cap T^{-1}Q_{i_1} \cap \cdots \cap T^{-N}Q_{i_N} \right )\gt 0 \} \\ \end{align} }[/math]

\begin{align} \bigvee_{n=0}^N T^{-n}Q & = \{Q_{i_0} \cap T^{-1}Q_{i_1} \cap \cdots \cap T^{-N}Q_{i_N} \\ & {} \qquad \mbox { where }i_\ell = 1,\ldots,k ,\ \ell=0,\ldots,N,\ \\ & {} \qquad \qquad \mu \left (Q_{i_0} \cap T^{-1}Q_{i_1} \cap \cdots \cap T^{-N}Q_{i_N} \right )>0 \} \\ \end{align}

which plays crucial role in the construction of the measure-theoretic entropy of a dynamical system.

它在动力系统的测量理论熵的构造中起着关键作用。

Measure-theoretic entropy

The entropy of a partition [math]\displaystyle{ \mathcal{Q} }[/math] is defined as^[2]^[3]

The entropy of a partition \mathcal{Q} is defined as

= = = 测量理论熵 = = = 一个分区数学{ q }的熵定义为

[math]\displaystyle{ H(\mathcal{Q})=-\sum_{Q \in \mathcal{Q}}\mu (Q) \log \mu(Q). }[/math]

H(\mathcal{Q})=-\sum_{Q \in \mathcal{Q}}\mu (Q) \log \mu(Q).

h (mathcal { q }) =-sum _ { q in mathcal { q } mu (q) log mu (q).

The measure-theoretic entropy of a dynamical system [math]\displaystyle{ (X, \mathcal{B}, T, \mu) }[/math] with respect to a partition Q = {Q₁, ..., Q_k} is then defined as

The measure-theoretic entropy of a dynamical system (X, \mathcal{B}, T, \mu) with respect to a partition Q = {Q1, ..., Qk} is then defined as

对于一个分区 q = { Q1，... ，Qk } ，动力系统(x，数学{ b } ，t，mu)的度量-理论熵定义为:

[math]\displaystyle{ h_\mu(T,\mathcal{Q}) = \lim_{N \rightarrow \infty} \frac{1}{N} H\left(\bigvee_{n=0}^N T^{-n}\mathcal{Q}\right). }[/math]

h_\mu(T,\mathcal{Q}) = \lim_{N \rightarrow \infty} \frac{1}{N} H\left(\bigvee_{n=0}^N T^{-n}\mathcal{Q}\right).

h _ mu (t，mathcal { q }) = lim _ { n 右行 infrc {1}{ n } h 左(bigvee _ { n = 0} ^ n t ^ {-n }数学{ q }右)。

Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system [math]\displaystyle{ (X, \mathcal{B},T,\mu) }[/math] is defined as

Finally, the Kolmogorov–Sinai metric or measure-theoretic entropy of a dynamical system (X, \mathcal{B},T,\mu) is defined as

最后，动力系统的 Kolmogorov-Sinai 度量或度量理论熵(x，mathcal { b } ，t，mu)被定义为

[math]\displaystyle{ h_\mu(T) = \sup_Q h_\mu(T,Q). }[/math]

h_\mu(T) = \sup_Q h_\mu(T,Q).

h _ mu (t) = sup _ q h _ mu (t，q).

where the supremum is taken over all finite measurable partitions. A theorem of Yakov Sinai in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process is log 2, since almost every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals [0, 1/2) and [1/2, 1]. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2ⁿx.

where the supremum is taken over all finite measurable partitions. A theorem of Yakov Sinai in 1959 shows that the supremum is actually obtained on partitions that are generators. Thus, for example, the entropy of the Bernoulli process is log 2, since almost every real number has a unique binary expansion. That is, one may partition the unit interval into the intervals [0, 1/2) and [1/2, 1]. Every real number x is either less than 1/2 or not; and likewise so is the fractional part of 2nx.

其中上确界占据了所有有限的可测分区。1959年 Yakov Sinai 的一个定理证明了上确界实际上是在生成元上得到的。因此，例如，伯努利过程的熵是 log 2，因为几乎每个实数都有唯一的二进制展开式。也就是说，可以将单位间隔划分为[0,1/2]和[1/2,1]。每个实数 x 要么小于1/2，要么不小于; 2nx 的小数部分也是如此。

If the space X is compact and endowed with a topology, or is a metric space, then the topological entropy may also be defined.

如果空间 x 是紧的并且具有拓扑结构，或者是度量空间，那么拓扑熵空间也可以被定义。

Classification and anti-classification theorems

One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let [math]\displaystyle{ (X, \mathcal{B}, \mu) }[/math] be a measure space, and let [math]\displaystyle{ U }[/math] be the set of all measure preserving systems [math]\displaystyle{ (X, \mathcal{B}, \mu, T) }[/math]. An isomorphism [math]\displaystyle{ S\sim T }[/math] of two transformations [math]\displaystyle{ S, T }[/math] defines an equivalence relation [math]\displaystyle{ \mathcal{R}\subset U\times U. }[/math] The goal is then to describe the relation [math]\displaystyle{ \mathcal{R} }[/math]. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms.^[4]^[5]

One of the primary activities in the study of measure-preserving systems is their classification according to their properties. That is, let (X, \mathcal{B}, \mu) be a measure space, and let U be the set of all measure preserving systems (X, \mathcal{B}, \mu, T). An isomorphism S\sim T of two transformations S, T defines an equivalence relation \mathcal{R}\subset U\times U. The goal is then to describe the relation \mathcal{R}. A number of classification theorems have been obtained; but quite interestingly, a number of anti-classification theorems have been found as well. The anti-classification theorems state that there are more than a countable number of isomorphism classes, and that a countable amount of information is not sufficient to classify isomorphisms.

= = 分类和反分类定理 = = 保测度系统研究的主要活动之一是根据它们的性质对它们进行分类。也就是说，设(x，数学{ b } ，mu)是一个测度空间，u 是所有测度保持系统(x，数学{ b } ，mu，t)的集合。变换的同构 s sim t，t 定义了一个等价关系数学子集{ r } u 乘以 u。然后目标是描述关系数学{ r }。已经获得了许多分类定理，但是相当有趣的是，还发现了许多反分类定理。反分类定理指出，同构类的可数个以上，可数信息量不足以对同构进行分类。

The first anti-classification theorem, due to Hjorth, states that if [math]\displaystyle{ U }[/math] is endowed with the weak topology, then the set [math]\displaystyle{ \mathcal{R} }[/math] is not a Borel set.^[6] There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type.^[7]

The first anti-classification theorem, due to Hjorth, states that if U is endowed with the weak topology, then the set \mathcal{R} is not a Borel set. There are a variety of other anti-classification results. For example, replacing isomorphism with Kakutani equivalence, it can be shown that there are uncountably many non-Kakutani equivalent ergodic measure-preserving transformations of each entropy type.

由 Hjorth 提出的第一个反分类定理指出，如果 u 被赋予弱拓扑，那么集合数学{ r }就不是 Borel 集。还有其他各种各样的反分类结果。例如，用 Kakutani 等价代替同构，可以证明每个熵类型都有不可数的非 Kakutani 等价的遍历测度保持变换。

These stand in contrast to the classification theorems. These include:

Ergodic measure-preserving transformations with a pure point spectrum have been classified.^[8]
Bernoulli shifts are classified by their metric entropy.^[9]^[10]^[11] See Ornstein theory for more.