遍历理论

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Ergodic theory (Greek: 脚本错误:没有“lang”这个模块。 脚本错误:没有“lang”这个模块。 "work", 脚本错误:没有“lang”这个模块。 脚本错误:没有“lang”这个模块。 "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.

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特性。在这种情况下,统计性质是指通过沿动力系统轨迹的各种函数的时间平均表示的性质。确定性动力系统的概念假定决定动力学的方程不包含任何随机扰动、噪声等。因此,我们所关心的统计数据是动力学的性质。


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

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,是基于测量理论的一般概念。它最初的发展是受到统计物理学问题的推动。


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.

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关注点是动力系统被允许长时间运行时的行为。这个方向的第一个结果是相空间庞加莱始态复现定理,它声称相空间的任何子集中的几乎所有点最终都会重新访问这个集合。庞加莱始态复现定理持有的系统是保守系统,因此所有遍历系统都是保守系统。


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.

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为精确的信息,它们断言,在一定条件下,函数沿轨迹的时间平均值几乎无处不在,并且与空间平均值有关。其中两个最重要的定理是 Birkhoff (1931)和 von Neumann 的定理,它们断言在每个轨迹上存在一个时间平均值。对于一类特殊的遍历系统,这个时间平均值对于几乎所有的初始点都是相同的: 从统计学的角度来说,进化了很长时间的系统“忘记”了它的初始状态。更强的性质,如混合和均匀分布,也得到了广泛的研究。


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.

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组成部分。动力系统的各种熵概念在遍历理论及其在随机过程中的应用中发挥了突出的作用。


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).

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).

遍历性和遍历假设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。遍历理论在数学其他领域的应用通常涉及到建立特殊类型系统的遍历性质。在几何学中,从负曲率 Riemann 曲面的 Eberhard Hopf 结果出发,利用遍历理论的方法研究了黎曼流形上的测地线流。马尔可夫链在21概率论的应用中形成了一个共同的上下文。遍历理论与傅里叶分析,李理论(表示论,代数群中的格)和数论(丢番图逼近理论,l 函数)有着丰富的联系。


Ergodic transformations

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.

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.

遍历理论经常涉及遍历变换。这种转换在给定的一组中起作用,其背后的直觉是它们彻底地“搅动”了这一组的元素(例如,如果一组是碗中的热燕麦片的量,如果一勺糖浆被倒入碗中,那么与燕麦片遍历转换的反向迭代将不允许糖浆留在燕麦片的一个局部分区域中,而是将糖浆均匀地分布在整个碗中。与此同时,这些迭代不会压缩或扩张燕麦片的任何部分: 它们保留了密度。)下面是正式的定义。


Let T : XX 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.

Let be a measure-preserving transformation on a measure space , with 1}}. Then is ergodic if for every in with E}}, either 0}} or 1}}.

设是测度空间上的一个保测度变换,其中有1}。然后是遍历 if for every in with e } ,or 0} or 1}。


Examples

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.

经典系统集合在相空间中的演化(上)。这些系统是位于一维势阱中的巨大粒子(红色曲线,下图)。最初的紧凑集成成为漩涡随着时间和“扩散”周围的相空间。然而,这不是遍历行为,因为系统不访问左侧势阱。


  • An irrational rotation of the circle R/Z, T: xx + 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.
  • 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.


Ergodic theorems

Let T: XX 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:

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 是测度空间(x,σ,μ)上的一个保测度变换,并且 f f 是 μ 可积函数,即 f 是 μ 可积函数。ƒ ∈ L1(μ).然后我们定义下面的平均值:


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

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

时间平均值: 这是 t 从初始点 x 开始迭代的平均值(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]

[math]\displaystyle{ \hat f(x) = \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x). }[/math]

< math > hat f (x) = lim _ { n right tarrow infty } ; frac { 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:

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

< blockquote > 空间平均: 如果 μ (x)是有限而非零的,我们可以考虑一下 f:


[math]\displaystyle{ \bar f =\frac 1{\mu(X)} \int f\,d\mu.\quad\text{ (For a probability space, } \mu(X)=1.) }[/math]

[math]\displaystyle{ \bar f =\frac 1{\mu(X)} \int f\,d\mu.\quad\text{ (For a probability space, } \mu(X)=1.) }[/math]

1{ mu (x)} int f,d mu. quad text {(对于一个概率空间,} 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.

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,时间平均数和空间平均数可能是不同的。但是如果变换是遍历的,测度是不变的,那么时间平均值几乎处处等于空间平均值。这就是著名的遍历定理,由于乔治·戴维·伯克霍夫的存在而以抽象的形式出现。(实际上,Birkhoff 的论文不考虑抽象的一般情形,而只考虑光滑流形上的微分方程所产生的动力系统的情形。)等分布定理是遍历定理的一个特殊情形,它专门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:

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:

更准确地说,点态或强遍历定理说明了 f/时间平均的定义中的极限几乎对每个 x 都存在,而且(几乎所有的定义)极限函数是可积的:


[math]\displaystyle{ \hat f \in L^1(\mu). \, }[/math]

[math]\displaystyle{ \hat f \in L^1(\mu). \, }[/math]

L ^ 1(mu)中的 f。,math


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

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

此外,f 是 t 不变的,也就是说


[math]\displaystyle{ \hat f \circ T= \hat f \, }[/math]

[math]\displaystyle{ \hat f \circ T= \hat f \, }[/math]

这个世界就是这样,这个世界


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

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

几乎处处成立,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]

[math]\displaystyle{ \int \hat f\, d\mu = \int f\, d\mu. }[/math]

数学,数学


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

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 是遍历的,那么 f/肯定是一个常量(几乎在任何地方) ,所以我们有这个


[math]\displaystyle{ \bar f = \hat f \, }[/math]

[math]\displaystyle{ \bar f = \hat f \, }[/math]

数学,数学


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

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

几乎到处都是。假设 μ (x)是有限且非零的,那么把第一个和最后一个索赔联系起来,我们就有了 μ (x)


[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]

[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]

1}{ n } sum { k = 0} ^ { n-1} f (t ^ kx) = 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.

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

几乎所有的 x,也就是说,除了一组测度0之外,所有的 x。


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

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

对遍历变换,时间平均几乎肯定等于空间平均。


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.

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,假设测量空间(x,σ,μ)模拟气体的粒子,并且让 f (x)表示位置 x 处的粒子的速度。然后逐点遍历定理表明,在给定的时间内,所有粒子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.

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

伯克霍夫定理的推广是金曼的次可加遍历定理。


Probabilistic formulation: Birkhoff–Khinchin theorem

Probabilistic formulation: Birkhoff–Khinchin theorem

= = = 概率公式: 伯克霍夫-钦钦定理 = = < !——这一部分的标题是“概率公式: Birkhoff-Khinchin 定理”(用短破折号而不是连字符) ,由六个重定向页面链接而成。-->


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

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

伯克霍夫-钦钦定理。E (| f) < ∞ ,t 是一个保测度图。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]

[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]

{ n }{ n } sum { k = 0} ^ { n-1} f (t ^ kx) = 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.

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 e (f | mathcal { c }) </math > 是给出 t 的不变集的 σ- 代数 < math > 数学 > 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:

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) : 特别地,如果 t 也是遍历的,那么 < math > mathcal { c } </math > 是平凡的 σ- 代数,因此概率为1:


[math]\displaystyle{ \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x)=E(f). }[/math]

[math]\displaystyle{ \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^{n-1} f(T^k x)=E(f). }[/math]

< math > lim _ { n right tarrow infty } ; frac {1}{ n } sum { k = 0} ^ { n-1} f (t ^ k x) = e (f) . </math >


Mean ergodic theorem

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

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

Von Neumann 的平均遍历定理在 Hilbert 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 |  = ψ} = ker(I − U).

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 Hilbert 空间 h 上的幺正算符算子; 更一般地说,等距线性算子(即对 h 中的所有 x,不一定满足‖ Ux ‖ = ‖ x ‖ ,或等价地,满足 u * u = i,但不一定是 UU * = i)。设 p 是{ ψ ∈ h | uψ = ψ } = ker (i-u)上的正交投影。


Then, for any x in H, we have:

Then, for any x in H, we have:

那么,对于 h 中的任意 x,我们有:


[math]\displaystyle{ \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^{n} x = P x, }[/math]

[math]\displaystyle{ \lim_{N \to \infty} {1 \over N} \sum_{n=0}^{N-1} U^{n} x = P x, }[/math]

< math > 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

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

这里的极限相对于 h 的平均值,换句话说,平均值的序列


[math]\displaystyle{ \frac{1}{N} \sum_{n=0}^{N-1} U^n }[/math]

[math]\displaystyle{ \frac{1}{N} \sum_{n=0}^{N-1} U^n }[/math]

< math > frac {1}{ n } sum _ { n = 0} ^ { N-1} u ^ n </math >


converges to P in the strong operator topology.

converges to P in the strong operator topology.

在强算子拓扑中收敛到 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:

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,h </math > 中的任何 < math > x 都承认分解为分别来自 < math > ker (I-U) </math > 和 < math > overline { operatorname { ran }(I-U)} </math > 的正交分解。前者在所有的部分和中都是不变的,因为数学的增长,而后者在裂项和中是不变的:


[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]

[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]

< math > 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

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

这个定理专门针对 Hilbert 空间 h 由测度空间上的 l < sup > 2 函数组成,而 u 是形式算子的情形


[math]\displaystyle{ Uf(x) = f(Tx) \, }[/math]

[math]\displaystyle{ Uf(x) = f(Tx) \, }[/math]

< math > 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.

where T is a measure-preserving endomorphism of X, thought of in applications as representing a time-step of a discrete dynamical system. 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 的保测度自同态,在应用中被认为是表示离散动力系统的时间步长。遍历定理然后断言,一个函数 f/足够大的平均行为是由 f/时间尺度的正交分量近似,这是时间不变的。


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

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

在遍历平均定理的另一种形式中,设 u < sub > t 是 h 上酉算子的强连续单参数群。然后是接线员


[math]\displaystyle{ \frac{1}{T}\int_0^T U_t\,dt }[/math]

[math]\displaystyle{ \frac{1}{T}\int_0^T U_t\,dt }[/math]

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.

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 t →∞的强算子拓扑中收敛。事实上,这个结果也推广到自反空间上压缩算子的强连续单参数半群的情形。


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).

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).

注: 当单位长度的复数被看作复平面上的幺正变换时(通过左乘法) ,可以给出遍历中值定理的一些直观。如果我们选择一个单位长度的复数(我们认为是 u) ,它的幂将填充圆。因为这个圆在0周围是对称的,所以 u 的幂的平均值会收敛到0也就说得通了。此外,0是 u 的唯一不动点,因此在不动点空间上的投影必须是零算符(这与刚才描述的极限一致)。


Convergence of the ergodic means in the Lp norms

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.

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.

设(x,σ,μ)是上述具有保测度变换 t 的概率空间,且设1≤ p ≤∞。关于 t 不变集的子 σ- 代数 σ < sub > t t ,条件期望是 Banach 空间 l < sup > p (x,σ,μ)到其闭子空间 l < sup > p (x,σ < sub > t ,μ)的一个线性投影算子 e < sub > t t t 。遍历平均作为 l < sup > p (x,σ,μ)上的线性算子也有单位算子范数; 作为 Birkhoff-Khinchin 定理的一个简单推论,在 l < sup > p 的强算子拓扑中,当1≤ p ≤∞时,在弱算子拓扑中,收敛到投影算子 e < sub > t  ;。如果 f/l < p ≤∞ ,那么 Wiener-Yoshida-Kakutani 遍历勒贝格控制收敛定理则表明 l < sup > p p 的遍历方法在 l < sup > p 中占优势,但是如果 f/l < sup > 1 ,遍历方法在 l < sup > p 中可能不等式。最后,如果认为 Zygmund 级 f/e 是可积的,那么 l < sup > 1 中的遍历均值甚至是占优的。


Sojourn time

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:

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:

设(x,σ,μ)是一个测度空间,使得 μ (x)是有限且非零的。在一个可测量的集合 a 中度过的时间称为逗留时间。遍历定理的一个直接结果是,在遍历系统中,a 的相对测度等于平均逗留时间:


[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]

[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]

< math > frac { mu (a)}{ mu (x)} = frac 1{ mu (x)} int chi _ a,d mu = lim { n right tarrow 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.

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

对于所有的 x,除了一组测度0,其中 χ < sub > a 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 = kiki−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.

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 that the initial point x is in A, so that k0 = 0.

可测集 a 的出现次数定义为集 k < sub > 1 ,k < sub > 2 ,k < sub > 3 ,... ,次数 k 使 t < sup > k (x)在 a 中,按递增次序排列。连续出现次数 r < sub > i = k i & minus; k < sub > i & minus; 1 之间的差值称为 a 的复发次数。遍历定理的另一个结果是 a 的平均递推时间与 a 的测度成反比,假设初始点 x 在 a 中,因此 k < sub > 0 = 0。


[math]\displaystyle{ \frac{R_1 + \cdots + R_n}{n} \rightarrow \frac{\mu(X)}{\mu(A)} \quad\text{(almost surely)} }[/math]

[math]\displaystyle{ \frac{R_1 + \cdots + R_n}{n} \rightarrow \frac{\mu(X)}{\mu(A)} \quad\text{(almost surely)} }[/math]

[ math > frac { r _ 1 + cdots + r _ n }{ n } right tarrow frac { mu (x)}{ mu (a)}} quad text {(几乎肯定)} </math >


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

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

(几乎可以肯定)也就是说,a 越小,回到它的时间就越长。


Ergodic flows on manifolds

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.

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.

1939年,Eberhard Hopf 证明了变负曲率紧致 Riemann 曲面和任意维常负曲率紧致流形上测地流的遍历性,尽管之前已经研究过一些特殊情况,如 Hadamard 台球(1898)和 arstin 台球(1924)。1952年,s. v. Fomin 和 i. m. Gelfand 描述了 Riemann 曲面上的测地流与 SL (2,r)上的单参数子群之间的关系。关于 Anosov 流的文章给出了 SL (2,r)和负曲率 Riemann 曲面上的遍历流的一个例子。这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半单李群 SO (n,1)中格的作用下的双曲空间的商。在黎曼对称空间上测地线流的遍历性被 f. i. Mautner 于1957年证明。1967年,D.v. 阿诺索夫和亚。证明了可变负截面曲率紧致流形上测地流的遍历性。半单李群齐性空间上齐次流的遍历性的一个简单判据是 Calvin c. Moore 在1966年给出的。这个研究领域的许多定理和结果都是刚性理论的典型例子。


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.

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 证明了紧致双曲面上的周期流是极小且遍历的。流动的独特遍历性是由哈里·弗斯腾伯格于1972年建立的。Ratner 定理给出了形式为 γg 的齐次空间上单幂流的遍历性的一个重要推广,其中 g 是李群,γ 是 g 中的格。


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.

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.

近20年来,在 Furstenberg 和 Margulis 的猜想的推动下,有许多工作试图找到一个类似于 Ratner 定理但是可对角化的行为的测度分类定理。一个重要的部分结果(用额外的正熵假设来解决这些猜想)被埃隆·林登施特劳斯证明了,并且他因此在2010年获得了菲尔兹奖。


See also


References

  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

  • 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

  • Leo Breiman, Probability. Original edition published by Addison–Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. . (See Chapter 6.)
  • Tim Bedford; Michael Keane; Caroline Series, eds. (1991), Ergodic theory, symbolic dynamics and hyperbolic spaces, Oxford University Press, ISBN 0-19-853390-X (A survey of topics in ergodic theory; with exercises.)
  • Karl Petersen. Ergodic Theory (Cambridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press. 1990.
  • Joseph M. Rosenblatt and Máté Weirdl, Pointwise ergodic theorems via harmonic analysis, (1993) appearing in Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference, (1995) Karl E. Petersen and Ibrahim A. Salama, eds., Cambridge University Press, Cambridge, . (An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of shift maps on the unit interval. Focuses on methods developed by Bourgain.)
  • Joseph D. Zund (2002), "George David Birkhoff and John von Neumann: A Question of Priority and the Ergodic Theorems, 1931–1932", Historia Mathematica, 29 (2): 138–156, doi:10.1006/hmat.2001.2338 (A detailed discussion about the priority of the discovery and publication of the ergodic theorems by Birkhoff and von Neumann, based on a letter of the latter to his friend Howard Percy Robertson.)
  • Andrzej Lasota, Michael C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. Second Edition, Springer, 1994.


External links

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