# K-S熵

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.

## 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

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

$\displaystyle{ (X, \mathcal{B}, \mu, T) }$
(X, \mathcal{B}, \mu, T)
(x，数学{ b } ，mu，t)

with the following structure:

with the following structure:

• $\displaystyle{ X }$ is a set,
• $\displaystyle{ \mathcal B }$ is a σ-algebra over $\displaystyle{ X }$,
• $\displaystyle{ \mu:\mathcal{B}\rightarrow[0,1] }$ is a probability measure, so that $\displaystyle{ \mu (X) = 1 }$, and $\displaystyle{ \mu(\varnothing) = 0 }$,
• $\displaystyle{ T:X \rightarrow X }$ is a measurable transformation which preserves the measure $\displaystyle{ \mu }$, i.e., $\displaystyle{ \forall A\in \mathcal{B}\;\; \mu(T^{-1}(A))=\mu(A) }$.
• 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 $\displaystyle{ \mu(T^{-1}(A))=\mu(A) }$ instead of the forward transformation $\displaystyle{ \mu(T(A))=\mu(A) }$. This can be understood in a fairly easy fashion. Consider a mapping $\displaystyle{ \mathcal{T} }$ of power sets:

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

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

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.)

There are two classification problems of interest. One, discussed below, fixes $\displaystyle{ (X, \mathcal{B}, \mu) }$ and asks about the isomorphism classes of a transformation map $\displaystyle{ T }$. The other, discussed in transfer operator, fixes $\displaystyle{ (X, \mathcal{B}) }$ and $\displaystyle{ T }$, and asks about maps $\displaystyle{ \mu }$ 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.

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.

In terms of physics, the measure-preserving dynamical system $\displaystyle{ (X, \mathcal{B}, \mu, T) }$ 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 $\displaystyle{ T }$ describes this stirring, mixing, etc. then the system $\displaystyle{ (X, \mathcal{B}, \mu, T) }$ 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 $\displaystyle{ \mu }$ 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.

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.

## 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 $\displaystyle{ w\times l\times h, }$ consisting of $\displaystyle{ N }$ atoms. A single atom in that box might be anywhere, having arbitrary velocity; it would be represented by a single point in $\displaystyle{ w\times l\times h\times \mathbb{R}^3. }$ A given collection of $\displaystyle{ N }$ atoms would then be a single point somewhere in the space $\displaystyle{ (w\times l\times h)^N \times \mathbb{R}^{3N}. }$ 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 $\displaystyle{ X }$ 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 $\displaystyle{ \mu }$ is given by the Maxwell–Boltzmann distribution. It is a product measure, in that if $\displaystyle{ p_i(x,y,z,v_x,v_y,v_z)\,d^3x\,d^3p }$ is the probability of atom $\displaystyle{ i }$ having position and velocity $\displaystyle{ x,y,z,v_x,v_y,v_z }$, then, for $\displaystyle{ N }$ atoms, the probability is the product of $\displaystyle{ N }$ 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 $\displaystyle{ \mathcal{O}\left(2^{-3N}\right). }$ 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.

The only reason that this is an "informal example" is because writing down the transition function $\displaystyle{ T }$ 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.

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.

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.

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.

## Examples

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

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

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);
• 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 Ts : XX parametrized by sZ (or R, or N ∪ {0}, or [0, +∞)), where each transformation Ts satisfies the same requirements as T above.[1] In particular, the transformations obey the rules:

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

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

## Homomorphisms

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

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 $\displaystyle{ (X, \mathcal{A}, \mu, T) }$ and $\displaystyle{ (Y, \mathcal{B}, \nu, S) }$. Then a mapping

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

$\displaystyle{ \varphi:X \to Y }$
\varphi:X \to 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:

1. The map $\displaystyle{ \varphi\ }$ is measurable.
2. For each $\displaystyle{ B \in \mathcal{B} }$, one has $\displaystyle{ \mu (\varphi^{-1}B) = \nu(B) }$.
3. For $\displaystyle{ \mu }$-almost all $\displaystyle{ x \in X }$, one has $\displaystyle{ \varphi(Tx) = S(\varphi x) }$.
1. The map \varphi\ is measurable.
2. For each B \in \mathcal{B}, one has \mu (\varphi^{-1}B) = \nu(B).
3. For \mu-almost all x \in X, one has \varphi(Tx) = S(\varphi x).
1. 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 $\displaystyle{ (Y, \mathcal{B}, \nu, S) }$ is then called a factor of $\displaystyle{ (X, \mathcal{A}, \mu, T) }$.

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

The map $\displaystyle{ \varphi\; }$ 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

$\displaystyle{ \psi:Y \to X }$
\psi:Y \to X
psi: y to x

that is also a homomorphism, which satisfies

that is also a homomorphism, which satisfies

1. for $\displaystyle{ \mu }$-almost all $\displaystyle{ x \in X }$, one has $\displaystyle{ x = \psi(\varphi x) }$;
2. for $\displaystyle{ \nu }$-almost all $\displaystyle{ y \in Y }$, one has $\displaystyle{ y = \varphi(\psi y) }$.
1. for \mu-almost all x \in X, one has x = \psi(\varphi x);
2. for \nu-almost all y \in Y, one has y = \varphi(\psi y).

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

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

## Generic points

A point xX 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 $\displaystyle{ (X, \mathcal{B}, T, \mu) }$, and let Q = {Q1, ..., Qk} be a partition of X into k measurable pair-wise disjoint pieces. Given a point xX, 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

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

$\displaystyle{ T^nx \in Q_{a_n}. }$
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.

## Operations on partitions

Given a partition Q = {Q1, ..., Qk} and a dynamical system $\displaystyle{ (X, \mathcal{B}, T, \mu) }$, 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

$\displaystyle{ T^{-1}Q = \{T^{-1}Q_1,\ldots,T^{-1}Q_k\}. }$
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, ..., Qk} and R = {R1, ..., Rm}, define their refinement as

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

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

\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} }

\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}

\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.

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

## Measure-theoretic entropy

The entropy of a partition $\displaystyle{ \mathcal{Q} }$ 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

$\displaystyle{ H(\mathcal{Q})=-\sum_{Q \in \mathcal{Q}}\mu (Q) \log \mu(Q). }$
H(\mathcal{Q})=-\sum_{Q \in \mathcal{Q}}\mu (Q) \log \mu(Q).
h (mathcal { q }) =-sum _ { q in mathcal { q } mu (q) log mu (q).

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

$\displaystyle{ h_\mu(T,\mathcal{Q}) = \lim_{N \rightarrow \infty} \frac{1}{N} H\left(\bigvee_{n=0}^N T^{-n}\mathcal{Q}\right). }$
h_\mu(T,\mathcal{Q}) = \lim_{N \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 $\displaystyle{ (X, \mathcal{B},T,\mu) }$ is defined as

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

$\displaystyle{ h_\mu(T) = \sup_Q h_\mu(T,Q). }$
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.

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

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

## 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 $\displaystyle{ (X, \mathcal{B}, \mu) }$ be a measure space, and let $\displaystyle{ U }$ be the set of all measure preserving systems $\displaystyle{ (X, \mathcal{B}, \mu, T) }$. An isomorphism $\displaystyle{ S\sim T }$ of two transformations $\displaystyle{ S, T }$ defines an equivalence relation $\displaystyle{ \mathcal{R}\subset U\times U. }$ The goal is then to describe the relation $\displaystyle{ \mathcal{R} }$. 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.[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 $\displaystyle{ U }$ is endowed with the weak topology, then the set $\displaystyle{ \mathcal{R} }$ 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.

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

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

• Ergodic measure-preserving transformations with a pure point spectrum have been classified.
• Bernoulli shifts are classified by their metric entropy. See Ornstein theory for more.

• 贝努利位移按度量熵分类。更多信息参见奥恩斯坦理论。

• Krylov–Bogolyubov theorem on the existence of invariant measures
• Poincaré recurrence theorem

# = = =

• Krylov-Bogolyubov 关于不变测度存在性的庞加莱始态复现定理定理

## References

1. Walters, Peter (2000). An Introduction to Ergodic Theory. Springer. ISBN 0-387-95152-0.
2. Sinai, Ya. G. (1959). "On the Notion of Entropy of a Dynamical System". Doklady Akademii Nauk SSSR. 124: 768–771.
3. 模板:Cite document
4. Foreman, M.; Weiss, B. (2019). "From Odometers to Circular Systems: A Global Structure Theorem". Journal of Modern Dynamics. 15: 345–423. arXiv:1703.07093. doi:10.3934/jmd.2019024. S2CID 119128525.
5. Foreman, M.; Weiss, B. (2017). "Measure preserving Diffeomorphisms of the Torus are unclassifiable". arXiv:1705.04414. {{cite journal}}: Cite journal requires |journal= (help)
6. Hjorth, G. (2001). "On invariants for measure preserving transformations" (PDF). Fund. Math. 169 (1): 51–84. doi:10.4064/FM169-1-2. S2CID 55619325.
7. Ornstein, D.; Rudolph, D.; Weiss, B. (1982). Equivalence of measure preserving transformations. Mem. American Mathematical Soc.. 37. ISBN 0-8218-2262-4.
8. Halmos, P.; von Neumann, J. (1942). "Operator methods in classical mechanics. II". Annals of Mathematics. (2). 43 (2): 332–350. doi:10.2307/1968872. JSTOR 1968872.
9. Sinai, Ya. (1962). "A weak isomorphism of transformations with invariant measure". Doklady Akademii Nauk SSSR. 147: 797–800.
10. Ornstein, D. (1970). "Bernoulli shifts with the same entropy are isomorphic". Advances in Mathematics. 4 (3): 337–352. doi:10.1016/0001-8708(70)90029-0.
11. Katok, A.; Hasselblatt, B. (1995). "Introduction to the modern theory of dynamical systems". Encyclopedia of Mathematics and its Applications. 54. Cambridge University Press.

• Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). (Provides expository introduction, with exercises, and extensive references.)
• Lai-Sang Young, "Entropy in Dynamical Systems" (pdf; ps), appearing as Chapter 16 in Entropy, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003).
• T. Schürmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A 28(17), page 5033, 1995. PDF-Document (gives a more involved example of measure-preserving dynamical system.)
• Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). (Provides expository introduction, with exercises, and extensive references.)
• Lai-Sang Young, "Entropy in Dynamical Systems" (pdf; ps), appearing as Chapter 16 in Entropy, Andreas Greven, Gerhard Keller, and Gerald Warnecke, eds. Princeton University Press, Princeton, NJ (2003).
• T. Schürmann and I. Hoffmann, The entropy of strange billiards inside n-simplexes. J. Phys. A 28(17), page 5033, 1995. PDF-Document (gives a more involved example of measure-preserving dynamical system.)

有限型遍历理论和子移位》 ，(1991) ，作为《遍历理论，符号动力学和双曲空间》第二章，Tim Bedford，Michael Keane and Caroline Series，Eds。牛津大学出版社，1991年。(提供说明性的介绍，包括练习和广泛的参考资料。)

• 黎生荣，《动力系统中的熵》(pdf; ps) ，载于《熵》第十六章，Andreas Greven，Gerhard Keller，and Gerald Warnecke，eds。普林斯顿大学出版社，普林斯顿，新泽西州(2003)。
• t · 舒尔曼和 i · 霍夫曼，n 单形中奇异台球的熵。J. Phys.A28(17) ，第5033页，1995。Pdf 文档(给出了一个更复杂的保测动力系统的例子)

Category:Dynamical systems Category:Entropy and information Category:Measure theory Category:Entropy Category:Information theory

范畴: 动力系统范畴: 熵与信息范畴: 度量论范畴: 熵范畴: 信息论

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