遍历理论

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Ergodic theory (Greek: "work", "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics.

遍历理论（希腊语："工作"、"方式"）是数学的一个分支，研究确定性动态系统的统计属性。在这种情况下，统计属性是指通过各种函数沿动态系统轨迹的时间平均数的行为来表达的属性。确定性动态系统的概念假定决定动态的方程不包含任何随机扰动、噪声等。因此，我们所关注的统计数据是动力学的属性。


Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics.

遍历理论，像概率论一样，是基于度量理论的一般概念。其最初的发展是由统计物理学的问题所激发的。


A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set. Systems for which the Poincaré recurrence theorem holds are conservative systems; thus all ergodic systems are conservative.

遍历理论的一个核心问题是动态系统被允许长期运行时的行为。这个方向的第一个结果是庞加莱递归定理，该定理声称相空间任何子集的几乎所有点最终都会重访该集。庞加莱递归定理成立的系统是保守系统；因此所有的遍历系统都是保守的。


More precise information is provided by various ergodic theorems which assert that, under certain conditions, the time average of a function along the trajectories exists almost everywhere and is related to the space average. Two of the most important theorems are those of Birkhoff (1931) and von Neumann which assert the existence of a time average along each trajectory. For the special class of ergodic systems, this time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such as mixing and equidistribution, have also been extensively studied.

各种遍历定理提供了更精确的信息，这些定理断言，在某些条件下，一个函数沿轨迹的时间平均值几乎到处存在，并与空间平均值相关。两个最重要的定理是Birkhoff (1931)和von Neumann的定理，它们断言沿每个轨迹存在一个时间平均。对于特殊类别的遍历系统来说，这个时间平均数对几乎所有的初始点都是一样的：从统计学上讲，进化了很长时间的系统会 "忘记 "它的初始状态。更强的属性，如混合和等分，也被广泛研究。


The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications to stochastic processes is played by the various notions of entropy for dynamical systems.

系统的度量分类问题是抽象遍历理论的另一个重要部分。遍历理论及其在随机过程中的应用中，动态系统的各种熵概念发挥了突出的作用。


The concepts of ergodicity and the ergodic hypothesis are central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature. Markov chains form a common context for applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic groups), and number theory (the theory of diophantine approximations, L-functions).

遍历性和遍历假说的概念是遍历理论应用的核心。其基本思想是，对于某些系统，其属性的时间平均值等于整个空间的平均值。遍历理论在数学的其他部分的应用通常涉及到为特殊类型的系统建立遍历属性。在几何学中，从Eberhard Hopf对负曲率的黎曼面的结果开始，遍历理论的方法被用来研究黎曼流形上的测地流。马尔科夫链构成了概率论中应用的一个共同背景。遍历理论与谐波分析、李氏理论（表示理论、代数群中的格子）和数论（二项式近似理论、L-函数）有着丰富的联系。


Ergodic transformations 遍历变换

Main article: Ergodicity

Ergodic theory is often concerned with ergodic transformations. The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set (e.g., if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in a local subregion of the oatmeal, but will distribute the syrup evenly throughout. At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density.) Here is the formal definition.

遍历理论通常与遍历变换有关。这种作用于给定集合的变换背后的直觉是，它们能彻底地 "搅拌 "该集合的元素（例如，如果该集合是碗中的一定数量的热燕麦片，如果一勺糖浆被投入碗中，那么燕麦片的遍历变换的逆运算将不允许糖浆停留在燕麦片的局部子区域，而是将糖浆均匀地分布在整个区域。同时，这些迭代不会压缩或扩张燕麦片的任何部分：它们保留了作为密度的量度）。下面是正式的定义。


Let T : X → X be a measure-preserving transformation on a measure space (X, Σ, μ), with μ(X) = 1. Then T is ergodic if for every E in Σ with T−1(E) = E, either μ(E) = 0 or μ(E) = 1.

设T：X→X是度量空间（X，Σ，μ）上的一个度量保全变换，μ（X）=1。那么，如果对于Σ中的每一个E，T-1(E)=E，μ(E)=0或者μ(E)=1，那么T就是遍历的。


Examples 实例

文件:Hamiltonian flow classical.gif
Evolution of an ensemble of classical systems in phase space (top). The systems are massive particles in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time and "spread around" phase space. This is however not ergodic behaviour since the systems do not visit the left-hand potential well.

Evolution of an ensemble of classical systems in phase space (top). The systems are massive particles in a one-dimensional potential well (red curve, lower figure). The initially compact ensemble becomes swirled up over time and "spread around" phase space. This is however not ergodic behaviour since the systems do not visit the left-hand potential well.

一组经典系统在相空间的演变（上图）。这些系统是一维势阱中的大量粒子（红色曲线，下图）。最初紧凑的集合随着时间的推移变得旋转起来，并在相空间中 "扩散"。然而，这并不是遍历行为，因为这些系统并没有访问左边的势阱。


  • An irrational rotation of the circle R/Z, T: x → x + θ, where θ is irrational, is ergodic. This transformation has even stronger properties of unique ergodicity, minimality, and equidistribution. By contrast, if θ = p/q is rational (in lowest terms) then T is periodic, with period q, and thus cannot be ergodic: for any interval I of length a, 0 < a < 1/q, its orbit under T (that is, the union of I, T(I), ..., Tq−1(I), which contains the image of I under any number of applications of T) is a T-invariant mod 0 set that is a union of q intervals of length a, hence it has measure qa strictly between 0 and 1.
  • 圆R/Z的无理旋转，T：x→x+θ，其中θ是无理的，是遍历性的。这种变换具有更强的独特的遍历性、最小性和等分性的特性。相比之下，如果θ=p/q是有理的（以最低条件计算），那么T是周期性的，周期为q，因此不可能是遍历性的：对于任何长度为a的区间I，0<a<1/q，其在T下的轨道（即I，T(I)，...，Tq-1(I)的联合，包含I在T的任何应用次数下的图像）是一个T不变的mod 0集合，是长度为a的q区间的联合，因此它有严格在0和1之间的量度qa。


  • Let G be a compact abelian group, μ the normalized Haar measure, and T a group automorphism of G. Let G* be the Pontryagin dual group, consisting of the continuous characters of G, and T* be the corresponding adjoint automorphism of G*. The automorphism T is ergodic if and only if the equality (T*)n(χ) = χ is possible only when n = 0 or χ is the trivial character of G. In particular, if G is the n-dimensional torus and the automorphism T is represented by a unimodular matrix A then T is ergodic if and only if no eigenvalue of A is a root of unity.
  • 让G是一个紧凑的非线性群，μ是归一化的Haar度量，T是G的一个群自变形。让G*是Pontryagin对偶群，由G的连续字符组成，T*是G*的相应相邻自变形。当且仅当(T*)n(χ)=χ的等式只有在n=0或χ是G的琐碎字符时才可能出现时，自动形态T才是勘误的。特别是，如果G是n维环形，自动形态T由单模矩阵A表示，那么当且仅当A的特征值没有是统一根时，T才是勘误的。


  • A Bernoulli shift is ergodic. More generally, ergodicity of the shift transformation associated with a sequence of i.i.d. random variables and some more general stationary processes follows from Kolmogorov's zero–one law.
  • 伯努利移位是遍历性的。更一般地说，与一连串的独立随机变量和一些更一般的静止过程相关的移位变换的遍历性是由Kolmogorov的零一法则决定的。


  • Ergodicity of a continuous dynamical system means that its trajectories "spread around" the phase space. A system with a compact phase space which has a non-constant first integral cannot be ergodic. This applies, in particular, to Hamiltonian systems with a first integral I functionally independent from the Hamilton function H and a compact level set X = {(p,q): H(p,q) = E} of constant energy. Liouville's theorem implies the existence of a finite invariant measure on X, but the dynamics of the system is constrained to the level sets of I on X, hence the system possesses invariant sets of positive but less than full measure. A property of continuous dynamical systems that is the opposite of ergodicity is complete integrability.
  • 连续动力系统的遍历性意味着其轨迹在相空间中 "扩散"。一个具有紧凑相空间的系统如果具有非常数的第一积分，就不可能是遍历性的。这尤其适用于具有独立于汉密尔顿函数H的第一积分I和紧凑水平集X={(p,q): H(p,q)=E}的恒定能量。Liouville定理意味着X上存在一个有限的不变度量，但系统的动力学被限制在X上I的水平集上，因此系统拥有正的但不完全的不变度量集。连续动力系统的一个与遍历性相反的属性是完全整数性。


Ergodic theorems 遍历定理

Let T: X → X be a measure-preserving transformation on a measure space (X, Σ, μ) and suppose ƒ is a μ-integrable function, i.e. ƒ ∈ L1(μ). Then we define the following averages:

设T：X→X是度量空间（X，Σ，μ）上的一个度量保全变换，并假设ƒ是一个μ不可变的函数，即ƒ∈L1（μ）。那么我们定义以下的平均数：


Time average: This is defined as the average (if it exists) over iterations of T starting from some initial point x:

时间平均：这被定义为从某个初始点x开始的T的迭代的平均值（如果它存在的话）：[math]\displaystyle{ \hat f(x) = \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x). }[/math]



Space average: If μ(X) is finite and nonzero, we can consider the space or phase average of ƒ:

空间平均：如果μ(X)是有限的和非零的，我们可以考虑ƒ的空间或相位平均：[math]\displaystyle{ \bar f =\frac 1{\mu(X)} \int f\,d\mu.\quad\text{ (For a probability space, } \mu(X)=1.) }[/math]



In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space average almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff. (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.

一般来说，时间平均和空间平均可能是不同的。但如果变换是遍历性的，并且度量是不变的，那么时间平均数几乎在任何地方都等于空间平均数。这就是著名的遍历定理，其抽象形式是由乔治-大卫-伯克霍夫提出的。(实际上，伯克霍夫的论文考虑的不是抽象的一般情况，而只是由光滑流形上的微分方程产生的动力系统的情况）。等分布定理是遍历定理的一个特例，具体处理单位区间上的概率分布。


More precisely, the pointwise or strong ergodic theorem states that the limit in the definition of the time average of ƒ exists for almost every x and that the (almost everywhere defined) limit function ƒ̂ is integrable:

更确切地说，点态或强遍历定理指出，几乎每一个x都存在ƒ的时间平均值定义中的极限，并且（几乎所有地方定义的）极限函数ƒ̂是可积的：[math]\displaystyle{ \hat f \in L^1(\mu). \, }[/math]


Furthermore, [math]\displaystyle{ \hat f }[/math] is T-invariant, that is to say

此外, [math]\displaystyle{ \hat f }[/math] 是T恒定的的，即是说[math]\displaystyle{ \hat f \circ T= \hat f \, }[/math]


holds almost everywhere, and if μ(X) is finite, then the normalization is the same:

几乎在任何地方都成立，如果μ(X)是有限的，那么归一化是相同的：[math]\displaystyle{ \int \hat f\, d\mu = \int f\, d\mu. }[/math]



In particular, if T is ergodic, then ƒ̂ must be a constant (almost everywhere), and so one has that

特别是，如果T是遍历性的，那么ƒ̂必须是一个常数（几乎在任何地方），因此我们可以看到 [math]\displaystyle{ \bar f = \hat f \, }[/math]


almost everywhere. Joining the first to the last claim and assuming that μ(X) is finite and nonzero, one has that

几乎无处不在。将第一个要求与最后一个要求结合起来，并假设μ(X)是有限的和非零的，我们可以得出[math]\displaystyle{ \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x) = \frac 1 {\mu(X)} \int f\,d\mu }[/math]



for almost all x, i.e., for all x except for a set of measure zero.

对于几乎所有的x，也就是说，对于所有的x，除了一个度量为零的集合之外。


For an ergodic transformation, the time average equals the space average almost surely.

对于一个遍历性变换，时间平均数几乎肯定等于空间平均数。


As an example, assume that the measure space (X, Σ, μ) models the particles of a gas as above, and let ƒ(x) denote the velocity of the particle at position x. Then the pointwise ergodic theorems says that the average velocity of all particles at some given time is equal to the average velocity of one particle over time.

作为一个例子，假设度量空间（X, Σ, μ）如上所述对气体的粒子进行建模，让ƒ(x)表示粒子在位置x处的速度，那么，点态遍历定理说，所有粒子在某个特定时间的平均速度等于一个粒子在一段时间内的平均速度。


A generalization of Birkhoff's theorem is Kingman's subadditive ergodic theorem.

伯克霍夫定理的一个概括是金曼的次加性遍历定理。


Probabilistic formulation: Birkhoff–Khinchin theorem 概率论的表述：Birkhoff-Khinchin定理

Birkhoff–Khinchin theorem. Let ƒ be measurable, E(|ƒ|) < ∞, and T be a measure-preserving map. Then with probability 1:

Birkhoff-Khinchin定理：假设ƒ是可测的，E(|ƒ|)< ∞，T是一个保测的地图。那么概率为1。[math]\displaystyle{ \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x)=E(f \mid \mathcal{C})(x), }[/math]

where [math]\displaystyle{ E(f|\mathcal{C}) }[/math] is the conditional expectation given the σ-algebra [math]\displaystyle{ \mathcal{C} }[/math] of invariant sets of T.

其中 [math]\displaystyle{ E(f|\mathcal{C}) }[/math] 是鉴于T的不变集的σ代数 [math]\displaystyle{ \mathcal{C} }[/math] 的条件期望。



Corollary (Pointwise Ergodic Theorem): In particular, if T is also ergodic, then [math]\displaystyle{ \mathcal{C} }[/math] is the trivial σ-algebra, and thus with probability 1:

推论（点态遍历定理）：特别是，如果T也是遍历的，那么 [math]\displaystyle{ \mathcal{C} }[/math] 是琐碎的σ代数，因此概率是1。 [math]\displaystyle{ \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x)=E(f). }[/math]


Mean ergodic theorem 平均遍历定理

Von Neumann's mean ergodic theorem, holds in Hilbert spaces.[1]

冯-诺依曼的平均遍历定理，在希尔伯特空间中成立。


Let U be a unitary operator on a Hilbert space H; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying ‖Ux‖ = ‖x‖ for all x in H, or equivalently, satisfying U*U = I, but not necessarily UU* = I). Let P be the orthogonal projection onto {ψ ∈ H | Uψ = ψ} = ker(I − U).

设U是希尔伯特空间H上的一个单元算子；更一般地说，是一个等距线性算子（即，对于H中的所有x，满足 "Ux"="x "的不一定是射影线性算子，或者等价地，满足U*U = I，但不一定是UU* = I）。设P为正交投影到{ψ∈H | Uψ = ψ}= ker(I - U)。


Then, for any x in H, we have:

那么，对于H中的任何x，我们有： [math]\displaystyle{ \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^{n} x = P x, }[/math]


where the limit is with respect to the norm on H. In other words, the sequence of averages

其中极限是关于H上的规范。 换句话说，平均数的序列 [math]\displaystyle{ \frac{1}{N} \sum_{n=0}^{N-1} U^n }[/math]


converges to P in the strong operator topology.

在强算子拓扑中收敛于P。


Indeed, it is not difficult to see that in this case any [math]\displaystyle{ x\in H }[/math] admits an orthogonal decomposition into parts from [math]\displaystyle{ \ker(I-U) }[/math] and [math]\displaystyle{ \overline{\operatorname{ran}(I-U)} }[/math] respectively. The former part is invariant in all the partial sums as [math]\displaystyle{ N }[/math] grows, while for the latter part, from the telescoping series one would have:

事实上，不难看出，在这种情况下，任何一个y[math]\displaystyle{ x\in H }[/math] 都可以被正交地分解为分别来自 [math]\displaystyle{ \ker(I-U) }[/math] 和 [math]\displaystyle{ \overline{\operatorname{ran}(I-U)} }[/math] 的部分。前一部分在所有部分和中是不变的，随着 [math]\displaystyle{ N }[/math] 增长，而对于后一部分，裂项和是不变的： [math]\displaystyle{ \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^n (I-U)=\lim_{N \to \infty} {1 \over N} (I-U^N)=0 }[/math]


This theorem specializes to the case in which the Hilbert space H consists of L2 functions on a measure space and U is an operator of the form

该定理适用于这样的情况：希尔伯特空间H由度量空间上的L2函数组成，U是一个形式的算子 [math]\displaystyle{ Uf(x) = f(Tx) \, }[/math]


where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system.[2] The ergodic theorem then asserts that the average behavior of a function ƒ over sufficiently large time-scales is approximated by the orthogonal component of ƒ which is time-invariant.

其中T是X的一个度量保全内构，在应用中被认为是代表离散动力系统的一个时间步。遍历定理认为，函数ƒ在足够大的时间尺度上的平均行为是由ƒ的正交分量近似的，而正交分量是时间不变的。


In another form of the mean ergodic theorem, let Ut be a strongly continuous one-parameter group of unitary operators on H. Then the operator

在平均遍历定理的另一种形式中，让Ut是H上一个强连续的单参数的单元算子组，然后算子 [math]\displaystyle{ \frac{1}{T}\int_0^T U_t\,dt }[/math]


converges in the strong operator topology as T → ∞. In fact, this result also extends to the case of strongly continuous one-parameter semigroup of contractive operators on a reflexive space.

在强算子拓扑中随着T→∞而收敛。事实上，这一结果也延伸到了反身空间上的强连续单参数收缩算子半群的情况。


Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of as U), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers of U will converge to 0. Also, 0 is the only fixed point of U, and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).

备注：通过考虑单位长度的复数被视为复平面上的单位变换（通过左乘法）的情况，可以形成对平均遍历定理的一些直观认识。如果我们选择一个单一的单位长度的复数（我们认为是U），很直观的是它的幂将填满这个圆。由于圆是围绕0对称的，所以U的幂的平均数将收敛到0是有道理的。而且，0是U的唯一固定点，所以对固定点空间的投影必须是零算子（这与刚才描述的极限一致）。


Convergence of the ergodic means in the Lp norms Lp规范中的遍历手段的收敛性

Let (X, Σ, μ) be as above a probability space with a measure preserving transformation T, and let 1 ≤ p ≤ ∞. The conditional expectation with respect to the sub-σ-algebra ΣT of the T-invariant sets is a linear projector ET of norm 1 of the Banach space Lp(X, Σ, μ) onto its closed subspace Lp(X, ΣT, μ) The latter may also be characterized as the space of all T-invariant Lp-functions on X. The ergodic means, as linear operators on Lp(X, Σ, μ) also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projector ET in the strong operator topology of Lp if 1 ≤ p ≤ ∞, and in the weak operator topology if p = ∞. More is true if 1 < p ≤ ∞ then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of ƒ ∈ Lp are dominated in Lp; however, if ƒ ∈ L1, the ergodic means may fail to be equidominated in Lp. Finally, if ƒ is assumed to be in the Zygmund class, that is |ƒ| log+(|ƒ|) is integrable, then the ergodic means are even dominated in L1.

让(X, Σ, μ)如上所述是一个具有度量保全变换T的概率空间，并让1≤p≤∞。关于T不变集的子σ代数ΣT的条件期望是Banach空间Lp(X, Σ, μ)对其封闭子空间Lp(X, ΣT, μ)的规范1的线性投影ET，后者也可以被描述为X上所有T不变的Lp函数的空间。遍历的意思是，作为Lp(X, Σ, μ)上的线性算子也有单位算子规范；而且，作为Birkhoff-Khinchin定理的一个简单结果，如果1≤p≤∞，在Lp的强算子拓扑中收敛于投影仪ET，如果p=∞，在弱算子拓扑中收敛于投影仪。更多的是如果1 < p ≤ ∞，那么Wiener-Yoshida-Kakutani遍历支配收敛定理指出，ƒ∈Lp的遍历手段在Lp中是支配的；但是，如果ƒ∈L1，遍历手段在Lp中可能无法被等效支配。最后，如果假定ƒ属于Zygmund类，即|ƒ| log+(|ƒ|)是可整数的，那么遍历手段在L1中甚至是支配的。


Sojourn time 逗留时间

Let (X, Σ, μ) be a measure space such that μ(X) is finite and nonzero. The time spent in a measurable set A is called the sojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative measure of A is equal to the mean sojourn time:

让(X, Σ, μ)是一个度量空间，使得μ(X)是有限的和非零的。在一个可测集A中所花费的时间被称为停留时间（sojourn time）。遍历定理的一个直接后果是，在一个遍历系统中，A的相对度量等于平均逗留时间。 [math]\displaystyle{ \frac{\mu(A)}{\mu(X)} = \frac 1{\mu(X)}\int \chi_A\, d\mu = \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} \chi_A(T^k x) }[/math]


for all x except for a set of measure zero, where χA is the indicator function of A.

对于所有的x，除了一个度量为零的集合，其中χA是A的指示函数。


The occurrence times of a measurable set A is defined as the set k1, k2, k3, ..., of times k such that Tk(x) is in A, sorted in increasing order. The differences between consecutive occurrence times Ri = ki − ki−1 are called the recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A, assuming模板:Clarify that the initial point x is in A, so that k0 = 0.

一个可测量集A的发生时间被定义为，使Tk(x)在A中的时间k1, k2, k3, ..., 的集合，按递增的顺序排序。遍历定理的另一个结果是，A的平均遍历时间与A的度量成反比，假设模板：澄清初始点x在A中，所以k0 = 0。 [math]\displaystyle{ \frac{R_1 + \cdots + R_n}{n} \rightarrow \frac{\mu(X)}{\mu(A)} \quad\text{(almost surely)} }[/math]



(See almost surely.) That is, the smaller A is, the longer it takes to return to it.

(请看几乎是肯定的。)也就是说，A越小，返回的时间就越长。


Ergodic flows on manifolds 流形上的极化流

The ergodicity of the geodesic flow on compact Riemann surfaces of variable negative curvature and on compact manifolds of constant negative curvature of any dimension was proved by Eberhard Hopf in 1939, although special cases had been studied earlier: see for example, Hadamard's billiards (1898) and Artin billiard (1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups on SL(2, R) was described in 1952 by S. V. Fomin and I. M. Gelfand. The article on Anosov flows provides an example of ergodic flows on SL(2, R) and on Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of the hyperbolic space by the action of a lattice in the semisimple Lie group SO(n,1). Ergodicity of the geodesic flow on Riemannian symmetric spaces was demonstrated by F. I. Mautner in 1957. In 1967 D. V. Anosov and Ya. G. Sinai proved ergodicity of the geodesic flow on compact manifolds of variable negative sectional curvature. A simple criterion for the ergodicity of a homogeneous flow on a homogeneous space of a semisimple Lie group was given by Calvin C. Moore in 1966. Many of the theorems and results from this area of study are typical of rigidity theory.

Eberhard Hopf在1939年证明了变负曲率的紧凑黎曼面和任何维度的恒定负曲率的紧凑流的测地流的遍历性，尽管在更早的时候已经研究过特殊情况：例如，见Hadamard的台球（1898）和Artin的台球（1924）。1952年，S. V. Fomin和I. M. Gelfand描述了黎曼面上的测地流和SL(2, R)上的单参数子群之间的关系。关于阿诺索夫流的文章提供了SL(2, R)和负曲率黎曼表面上的遍历流的例子。那里描述的大部分发展都可以推广到双曲流形，因为它们可以被看作是双曲空间的商，由半纯李群SO(n,1)中的格子作用。Riemannian对称空间上测地流的遍历性由F. 毛特纳在1957年证明了黎曼尼对称空间上测地流的遍历性。1967年，D. V. Anosov和Ya. G. Sinai证明了测地流在可变负截面曲率的紧凑流形上的遍历性。1966年，卡尔文-C-摩尔给出了半纯李群的同质空间上同质流的啮合性的简单标准。这个研究领域的许多定理和结果是典型的刚度理论。


In the 1930s G. A. Hedlund proved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established by Hillel Furstenberg in 1972. Ratner's theorems provide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ \ G, where G is a Lie group and Γ is a lattice in G.

在20世纪30年代，G. A. Hedlund证明了紧凑双曲表面上的角循环流是最小和无规律的。1972年，Hillel Furstenberg建立了该流的唯一遍历性。拉特纳的定理为Γ\G形式的同质空间上的单势流提供了一个重要的概括，其中G是一个李群，Γ是G中的一个格子。


In the last 20 years, there have been many works trying to find a measure-classification theorem similar to Ratner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis. An important partial result (solving those conjectures with an extra assumption of positive entropy) was proved by Elon Lindenstrauss, and he was awarded the Fields medal in 2010 for this result.

在过去的20年里，有许多工作试图找到一个类似于拉特纳定理的措施分类定理，但对于可对角线化的行动，其动机是弗斯滕伯格和马古利斯的猜想。一个重要的部分结果（用一个额外的正熵假设来解决这些猜想）被Elon Lindenstrauss证明了，他因为这个结果在2010年获得了菲尔兹奖。


See also 另见


References 参考文献

  1. ↑ Reed, Michael; Simon, Barry (1980). Functional Analysis. Methods of Modern Mathematical Physics. 1 (Rev. ed.). Academic Press. ISBN 0-12-585050-6. 
  2. ↑ 模板:Harv


Historical references 历史参考文献

  • Birkhoff, George David (1931), "Proof of the ergodic theorem", Proc. Natl. Acad. Sci. USA, 17 (12): 656–660, Bibcode:1931PNAS...17..656B, doi:10.1073/pnas.17.12.656, PMC 1076138, PMID 16577406.
  • Hopf, Eberhard (1939), "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Leipzig Ber. Verhandl. Sächs. Akad. Wiss., 91: 261–304.


Modern references 现代参考文献

取自“https://wiki.swarma.org/index.php?title=遍历理论&oldid=33553”