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Ergodicity and Mixing Properties 225 Ergodicity and Mixing Properties 225 Ergodicity and Mixing Properties average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposi- ANTHONY QUAS tion gives a precise description of the manner in which Department of Mathematics and Statistics, a system can be split into ergodic components. University of Victoria, Victoria, Canada A related (stronger) property of a measure-preserv- ing transformation is mixing. Here one is investigating Article Outline the correlation between the state of the system at different times. The system is mixing if the states are asymptotically Glossary independent: as the times between the measurements in- Definition of the Subject crease to infinity, the observed values of the measurements Introduction at those times become independent. Basics and Examples Ergodicity Ergodic Decomposition Introduction Mixing The term ergodic was introduced by Boltzmann [8,9]in Hyperbolicity and Decay of Correlations his work on statistical mechanics, where he was study- Future Directions ing Hamiltonian systems with large numbers of particles. Bibliography The system is described at any time by a point of phase space,asubsetofR6N where N is the number of parti- Glossary cles. The configuration describes the 3-dimensional posi- Bernoulli shift Mathematical abstraction of the scenario tion and velocity of each of the N particles. It has long been in statistics or probability in which one performs re- known that the Hamiltonian (i. e. the overall energy of the peated independent identical experiments. system) is invariant over time in these systems. Thus, given Markov chain A probability model describing a sequence a starting configuration, all future configurations as the of observations made at regularly spaced time intervals system evolves lie on the same energy surface as the initial such that at each time, the probability distribution of one. the subsequent observation depends only on the cur- Boltzmann’s ergodic hypothesis was that the trajectory rent observation and not on prior observations. of the configuration in phase space would fill out the en- Measure-preserving transformation Amapfromamea- tire energy surface. The term ergodic is thus an amalgama- sure space to itself such that for each measurable sub- tion of the Greek words for work and path. This hypothe- set of the space, it has the same measure as its inverse sis then allowed Boltzmann to conclude that the long-term image under the map. average of a quantity as the system evolves would be equal Measure-theoretic entropy A non-negative (possibly in- to its average value over the phase space. finite) real number describing the complexity of a mea- Subsequently, it was realized that this hypothesis is sure-preserving transformation. rarely satisfied. The ergodic hypothesis was replaced in Product transformation Given a pair of measure-pre- 1911 by the quasi-ergodic hypothesis of the Ehrenfests [17] serving transformations: T of X and S of Y,theprod- which stated instead that each trajectory is dense in the uct transformation is the map of X Y given by energy surface, rather than filling out the entire energy (T S)(x; y) D (T(x); S(y)). surface. The modern notion of ergodicity (to be defined below) is due to Birkhoff and Smith [7]. Koopman [44] suggested studying a measure-preserving transformation Definition of the Subject by means of the associated isometry on Hilbert space, 2 2 Many physical phenomena in equilibrium can be modeled UT : L (X) ! L (X)definedbyUT ( f ) D f ı T.This as measure-preserving transformations. Ergodic theory is point of view was used by von Neumann [91] in his proof the abstract study of these transformations, dealing in par- of the mean ergodic theorem. This was followed closely ticular with their long term average behavior. by Birkhoff [6] proving the pointwise ergodic theorem. One of the basic steps in analyzing a measure-preserv- An ergodic measure-preserving transformation enjoys the ing transformation is to break it down into its simplest property that Boltzmann first intended to deduce from his possible components. These simplest components are its hypothesis: that long-term averages of an observable quan- ergodic components, and on each of these components, tity coincide with the integral of that quantity over the the system enjoys the ergodic property: the long-term time phase space. 226 Ergodicity and Mixing Properties These theorems allow one to deduce a form of in- a Lebesgue space (that is, the space together with its com- dependence on the average: given two sets of configura- pleted -algebra agrees up to a measure-preserving bijec- tions A and B, one can consider the volume of the phase tion with the unit interval with Lebesgue measure and the space consisting of points that are in A at time 0 and in B usual -algebra of Lebesgue measurable sets). Although at time t. In an ergodic measure-preserving transforma- this sounds like a strong restriction, in practice it is barely tion, if one computes the average of the volumes of these a restriction at all, as almost all of the spaces that appear regions over time, the ergodic theorems mentioned above in the theory (and all of those that appear in this article) allow one to deduce that the limit is simply the product of turn out to be Lebesgue spaces. For a detailed treatment the volume of A and the volume of B. This is the weak- of the theory of Lebesgue spaces, the reader is referred est mixing-type property. In this article, we will outline to Rudolph’s book [76]. The reader is referred also to the a rather full range of mixing properties with ergodicity at chapter on Measure Preserving Systems. the weakest end and the Bernoulli property at the strongest While many of the definitions that we shall present are end. valid for both invertible and non-invertible measure-pre- We will set out in some detail the various mixing prop- serving transformations, the strongest mixing conditions erties, basing our study on a number of concrete examples are most useful in the case of invertible transformations. sitting at various points of this hierarchy. Many of the mix- It will be helpful to present a selection of simple exam- ing properties may be characterized in terms of the Koop- ples, relative to which we will be able to explore ergodicity man operators operators mentioned above (i. e. they are and the various notions of mixing. These examples and the spectral properties), but we will see that the strongest mix- lemmas necessary to show that they are measure-preserv- ing properties are not spectral in nature. ing transformations as claimed may be found in the books We shall also see that there are connections between of Petersen [64], Rudolph [76]andWalters[92]. More de- the range of mixing properties that we discuss and mea- tails on these examples can also be found in the chapter on sure-theoretic entropy. In measure-preserving transfor- ErgodicTheory:BasicExamplesandConstructions. mations that arise in practice, there is a correlation be- Example 1 (Rotation on the circle) Let ˛ 2 R.Let tween strong mixing properties and positive entropy, al- R˛ :[0; 1) ! [0; 1) be defined by R˛(x) D x C ˛ mod 1. though many of these properties are logically independent. It is straightforward to verify that R˛ preserves the re- One important issue for which many questions remain striction of Lebesgue measure to [0; 1) (it is sufficient to open is that of higher-order mixing. Here, the issue is if 1 check that (R˛ (J)) D (J) for an interval J). instead of asking that the observations at two times sep- arated by a large time T be approximately independent, Example 2 (Doubling Map) Let M2 :[0; 1) ! [0; 1) be one asks whether if one makes observations at more times, defined by M2(x) D 2x mod 1. Again, Lebesgue measure each pair suitably separated, the results can be expected is invariant under M2 (to see this, one observes that for an 1 to be approximately independent. This issue has an ana- interval J, M2 (J) consists of two intervals, each of half the logue in probability theory, where it is well-known that it length of J). This may be generalized in the obvious way to is possible to have a collection of random variables that are amapMk for any integer k 2. pairwise independent, but not mutually independent. Example 3 (Interval Exchange Transformation) The class of interval exchange transformations was introduced by Sinai [85]. An interval exchange transformation is the map Basics and Examples obtained by cutting the interval into a finite number of In this article, except where otherwise stated, the measure- pieces and permuting them in such a way that the result- preserving transformations that we consider will be de- ing map is invertible, and restricted to each interval is an fined on probability spaces. order-preserving isometry. More specifically, given a measurable space (X; B)and More formally, one takes a sequence of positive B a probability measure defined on ,ameasure-preserv- lengths `1;`2;:::;`k summing to 1 andP a permutation B B ing transformation of (X; ;)isa -measurable map Pof f1;:::;kg and defines ai D j<i `j and bi D 1 B T : X ! X such that (T B) D (B)forallB 2 . (j)<(i) `j (again with b0 D 0). The interval exchange While this definition makes sense for arbitrary mea- transformation defined by (`1;:::;`k )and is the map sures, not simply probability measures, most of the re- T :[0; 1) ! [0; 1) defined by Tj[ai ;aiC1)(x) D xC(bi ai ). sults and definitions below only make sense in the proba- It is straightforward to check that any such interval ex- bility measure case.
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