Ergodic Theory

An overview of some of the more advanced topics we hope to pursue in a subsequent volume can be found in the lecture notes of Einsiedler and Lindenstrauss [80] in the Clay proceedings of the Pisa Summer School. A convenient summary of the measure-theoretic background can be found in the work of Royden [320] or of Parthasarathy [280].

Leitfaden

A minimal path to a uniform distribution of horocyclic orbits on SL2(Z)\SL2(R) would involve discussions of ergodicity from Chap.

Motivation

Examples of Ergodic Behavior
Equidistribution for Polynomials
Szemer´ edi’s Theorem
Szemer´ edi). Any subset of the integers with positive upper Banach density contains arbitrarily long arithmetic progressions

Indeﬁnite Quadratic Forms and Oppenheim’s Conjecture
Littlewood’s Conjecture
Integral Quadratic Forms
Dynamics on Homogeneous Spaces
An Overview of Ergodic Theory

7.3, and Furstenberg's proof of the necessary generalization of the Poincarian iteration will be presented in Sec. The proof of the theorem involves understanding the behavior of the orbits for the operation of the subgroup SO(2,1) SL3(R) at the points x ∈ SL3(Z)\SL3(R) (the space of right cosets of SL3(Z) in SL3 (R)); these can be thought of as arrays of the form xSO(2,1).

Notes to Chap. 1

The first published proof is that of Lagrange in 1770; a standard proof can be found in [87, Sect. 6) (Page 9). 8) For some of the statements made here, one must actually assume that Γ is lattice; see Sect.

Ergodicity, Recurrence and Mixing

Measure-Preserving Transformations

Note that the measure-preserving property cannot be seen by studying forward iterations: if I is a small interval, then T2(I) is an interval∗ of total length 2(b−a). One of the ways in which a measure-preserving transformation can be studied is through its induced action on some natural space of functions.

Fig. 2.1 The pre-image of [a, b) under the circle-doubling map

Recurrence

In this case, this means that almost every orbit of such a dynamical system returns infinitely often near its starting point (see Exercise 2.2.3(a)). A much deeper property that a dynamical system may have is that almost every orbit returns infinitely often close to almost every point, and this property is addressed in Section.2.3 (specifically in Proposition2.14).

Ergodicity

Ergodicity (indecomposability in the sense of goal theory) is a universal property of goal-preserving transformations in the sense that every goal-preserving transformation decomposes into ergodic components. Find a pair of ergodic measure-preserving systems (X,BX, μ, T) and (Y,BY, ν, S) for which T×S is not ergodic with respect to the product measure μ×ν.

Associated Unitary Operators

Then T is ergodic with respect to the Haar measuremX if and only if the identity χ(Tnx) =χ(x) for some n >0 and character χ ∈ X implies that χ is the trivial character with χ(x) = 1 for allx∈X. The transformation TA is ergodic if and only if no eigenvalue of A is a root of unity.

The Mean Ergodic Theorem

Using the mean ergodic theorem (Theorem 2.21) we know that for anyg∈L∞μ ⊆L2μ the ergodic meansAgN inL2μ converge to someg ∈L2μ. In other words, the ergodic means form a Cauchy series in L1μ, and thus have a limit f∈L1μ according to the Riesz-Fischer theorem (Theorem A.23).

Pointwise Ergodic Theorem

The Maximal Ergodic Theorem
Maximal Ergodic Theorem via Maximal Inequality
Maximal Ergodic Theorem via a Covering Lemma
The Pointwise Ergodic Theorem
Two Proofs of the Pointwise Ergodic Theorem

In this subsection, we use covering properties of intervals inZ to establish a version of the maximum ergodic theorem (Theorem 2.24). As with the maximum ergodic theorem (Theorem 2.24), we will give two proofs(24) of the pointwise ergodic theorem.

Strong-Mixing and Weak-Mixing

This seems to be asking too much (see Exercise 2.7.1), but asking that T−nB and A become asymptotically independent leads to the following non-trivial definition. Clearly, some gauge-preserving systems allow many sets to become asymptotically independent as they move apart in time (that is, under iteration), leading to the following natural definition due to Rokhlin [316]. It turns out that this is also true for measure-preserving transformations—there are weakly mixing transformations that do not mix(27).

Proof of Weak-Mixing Equivalences

Continuous Spectrum and Weak-Mixing

Due to the characterization in (2.31) and the ergodicity of S, the expression on the right in (2.34) converges to k. Assume that T×T is not ergodic, i.e. there exists a non-constant function f ∈ L2μ2(X×X) which is almost everywhere invariant under T×T. It follows that UT bounded to Vλ is a non-trivial linear mapping of a finite-dimensional linear space and thus has a non-trivial eigenvector.

Induced Transformations

The induced transformation TA is the map defined almost everywhere on the lower floor by sending each point to the point obtained by going through all the floors above it and returning to A. Poincar'e iteration (Theorem 2.11 ) say that for any measure-preserving system (X,B, μ, T) and setA of positive measure, almost every point on the ground floor of the associated Kakutani skyscraper returns to the ground floor at point . Ergodicity reinforces this statement to say that almost every point of the entire spaceX lies on some floor of the skyscraper.

Notes to Chap. 2

It is a special example of the ergodic mean theorem for Banach spaces, due to Kakutani and Yosida. Bishop's work [36] included a form of the ergodic theorem and Spitters [348] found constructive characterizations of the ergodic theorem. There are many proofs of the ergodic point theorem; besides that of Birkhoﬀ [33] there is a more elementary (albeit complicated) argument due to Katznelson and Weiss [186], motivated by a paper by Kamae [177].

Continued Fractions

Elementary Properties

The following simple lemma is crucial for many of the basic properties of the continued fraction expansion. It follows that a(pn−qnu) andb(pn−1− qn−1u) are of the same sign, hence the fact that. The argument used in the proof of Lemma3.4 also suggests a way to find the continued fractional expansion of a given irrational number lu∈RQ.

The Continued Fraction Map and the Gauss MeasureMeasure

This gives a description of the continued fraction map as a shift map: the list of digits in the continued fraction expansion of x ∈ [0,1]Q defines a unique element of NN, and the diagram. In Corollary3.8, we will draw some easy consequences(41) of ergodicity for the Gaussian measure μ in terms of properties of the continued fraction expansion for almost every real number. We will use the ergodicity of the Gaussian map in Corollary 3.8 to derive statements about the digits in the continued fraction expansion of a typical real number.

Badly Approximable Numbers

Lagrange’s Theorem

In the next section we generalize this example to show that all quadratic irrationals are badly approximable. The periodicity of the continued expansion of the fraction seen in Example 3.11 is a general property of quadratics. Since a quadratic polynomial has only two zeros, and un0, un1, un2 are all zeros of the same polynomial, we see that two of them coincide, so the continued expansion of the fraction is definitely periodic.

Invertible Extension of the Continued Fraction Map

More precisely, there is a countable union N of lines and curves in Y with the property that T|YN :YN →YN is a bijection preserving the Lebesgue measure. Since this is a Lebesgue zero set, it suffices to show that T|An:An→Bn is a bijection for every1, for then. The invertible extension, which preserves the Lebesgue measure, provides an alternative proof that the Gaussian measure is invariant and provides an explanation of where it might come from.

Fig. 3.2 The Gauss map is a bijection between Y and Y , sending the subset A n ⊆ Y to the subset B n ⊆ Y for each n 1

Notes to Chap. 3

Exercise 3.3.1 shows that we can take M5 = 4, and the question is raised in [259] whether there is a tighter limit that allows Md to be taken equal to 2 for all. 46) (Page 91) Liouville's theorem on the Diophantine approximation; there are several important results that bear his name) marked the beginning of a series of important advances in the Diophantine approximation, attempting to sharpen the lower bound. The assertion that for every algebraic number u of degree d there exists a constant c(u) so that for all rationals p/q we have |u−p/q| >.

Invariant Measures for Continuous Maps

Existence of Invariant Measures

In general it is difficult to identify measures with specific properties, but ergodic measures are easily characterized in terms of the geometry of the space of invariant measures. This collection of measures is an inconceivably small subset of the set of all ergodic measures—there is no hope of describing them all. Some of the problems for this section use the topological analogue of Definition 2.7, which will be used later.

Fig. 4.1 The North-South map on the circle; for z = 2i, T n z → 0 as n → ∞

Ergodic Decomposition

As we saw in Exercise 4.1.2, the set of ergodic measures is in general not a closed subset of the set of invariant measures. The existence of the ergodic decomposition is one of the reasons that ergodicity is such a powerful tool: whatever property is preserved by the integration in Theorem 4.8(2) that holds for ergodic systems holds for any measure-preserving transformation. Show that a topologically dynamic system is in general not an incoherent association of closed topologically ergodic subsystems.

Unique Ergodicity

1) ⇐⇒ (2): If T is uniquely ergodic and μ is the only T-invariant probability measure on X, then μ must be ergodic by Theorem 4.4. The equivalence of (1) and (3) in Theorem 4.10 first appeared in the paper of Kryloff and Bogoliouboff [214] in the context of unique ergodic flows. 1)Rg is uniquely ergodic (with unique invariant measure mX, Haar measure onX). 2) Rg is ergodic with respect to mX.

Measure Rigidity and Equidistribution

Equidistribution on the Interval
Equidistribution and Generic Points
Equidistribution for Irrational Polynomials

A more intuitive formulation (developed in Lemma 4.17) of the even distribution requires that the terms of the sequence fall in an interval with the correct frequency, just as the ergodic point theorem (Theorem 2.30) states that almost every orbit under an ergodic transformation falls into a measurable group with the appropriate frequency. 1) The sequence (xn) is uniformly distributed. More generally, Theorem 4.10 shows that the orbits of any circle transformation for which the Lebesgue measure is the unique invariant measure are uniformly distributed. It follows that the values of each of these polynomials are equal, so the values of the original polynomial are distributed modulo 1 by induction.

Fig. 4.2 The function χ [a,b] and the approximations f − (dots) and f + (dashes)

Notes to Chap. 4

Among the many extensions and modifications of this important result, Bellow and Furstenberg [22], Hansel and Raoult [140] and Denker [69] provided different proofs; Jakobs [164] and Denker and Eberlein [70] extended the result to flow; Lind and Thouvenot [231] showed that any finite entropy ergodic transformation is isomorphic to a homeomorphism of the torusT2-preserving Lebesgue measure;. Furstenberg and Weiss [109] showed that there is also a topological analogue of the ergodic multiple recurrence theorem (Theorem 7.4): if (X, T) is minimal and U⊆X is open and nonempty, then for any >1 there are somen1 with. 52)(Page110). Extensive reviews of this theory in three different decades can be found in the monographs of Kuipers and Niederreiter [215], Hlawka [154], and Drmota and Tichy [75].

Conditional Measures and Algebras

Conditional Expectation

The first step is existence, and we will give two different proofs that there exists a map E(·A) with the properties stated in (1). Equation (5.1) includes only functions that are continuous in L1, so there is a continuous extension for all L1 that still satisfies (5.1). Anyg∈L∞(X,A, μ) can be approximated by simple functions, so the general case follows from the continuity of the conditional expectation operator, which in turn is a consequence of inequality (5.2).

Martingales

Evidence. Assume thatf 0 (possibly replace f|f|, so thatμ(E) does not become smaller). In the decreasing case of a finite number of σ-algebras, we can reverse the order of the σ-algebras, since the statement we want to prove is independent of the order.) It follows that. The proof is in some respects similar to the proofs of the ergodic theorems (Theorems 2.21 and 2.30). To give an example of how it is sometimes possible to decompose functions in a way that mimics the orthogonal decomposition available in Hilbert space, we now do the same part of the proof avoiding L2.

Conditional Measures

While (5.6) looks simpler than (5.5), arriving at it requires constructing a quotient space and a quotient measure (see Section 5.4). Using dominant convergence, monotone convergence, and the monotone class theorem (Theorems A.18, A.16, and A.4), as in the proof of Theorem 5.14 on page 138, we can alternately extend the first and fourth properties to allf ∈C(X ), allf =χBfor any open set, any closed set, any Gδ, any Fσ, any Borel set and finally for any ∈L∞(X). Theorem 5.14(3) and the more geometrical discussion above emphasize the importance of the countably generated hypothesis on the σ-algebra A, since in this case the conditional measures μAx can be associated with the atoms of [x]A.

Algebras and Maps

If φ:X →Y is a Borel-measurable map between Borel subsets of compact metric spaces, then the induced mapφ∗:M(X)→M(Y) is Borel-measurable. In the next result, we will work with measurability on Borel subsets of compact metric spaces, without reference to a particular measure. For this, we start with the additional assumption that Z=Z is a compact metric space, and show later that this requirement can be removed.

Fig. 5.3 Each y ∈ Y determines an atom φ −1 (y) and its conditional measure ν y

Notes to Chap. 5

Factors and Joinings

The Ergodic Theorem and Decomposition Revisited
Invariant Algebras and Factor Maps
The Set of Joinings
Kronecker Systems
Constructing Joinings

In one direction, the connection is clear: given the factorial mapping, it is easy to construct an invariant σ-algebra, and we will do so in Lemma 6.4 (the proof of this is Exercise 6.2.1). Then there exists a measure-preserving system (Y,BY, ν, S) on the Borel probability space and a factor mappingφ:X →Y with A = φ−1BY (mod μ). By Lemma B.8, the space CC(X) is separable, so L2μ is separable, since the inclusion CC(X)→L2μ is connected to the dense image.

Fig. 6.1 Constructing the measurable map φ

Notes to Chap. 6

Furstenberg’s Proof of Szemer´ edi’s Theorem

Van der Waerden
Multiple Recurrence

Reduction to an Invertible System
Reduction to Borel Probability Spaces
Reduction to an Ergodic System

Furstenberg Correspondence Principle
An Instance of Polynomial Recurrence

Three immediate simplifications can be made to prove that all measure-preserving systems are SZ; the first two are somewhat technical, in that neither is necessary for the setting in which Theorem 7.5 is used to prove Szemer´edi's theorem (Theorem 1.5). In this short section, we show how Szemer´edi's theorem (Theorem 1.5) follows from the multiple repetition result in Theorem 7.5. 7.3, the Furstenberg correspondence principle can be used to show that Theorem 7.9 is a consequence of the following dynamic result due to(67) Furstenberg.