prelim (Pollicott, Yuri - Dynamical Systems and Ergodic Theory)

PRELIMINARIES1. Conventions. The book is divided into 16 chapters, each subdividedinto sections numbered in order (e.g. chapter 12, section 3 is numbered 12.3).Within each chapter results (Theorems, Propositions or Lemmas) are labelled by the chapter and then the order of occurrence (e.g. the fifth resultin chapter 3 is Proposition 3.5). The exceptions to this rule are: sublemmas which are presented within the context of the proof of a more importantresult (e.g. the proof of Theorem 2.2 contains Sublemmas 2.2.1 and 2.2.2);and corollaries (the corollary to Theorem 5.5 is Corollary 5.5.1).We denote the end of a proof by .Finally, equations are numbered by the chapter and their order of occurrence (e.g.

the fourth equation in chapter 5 is labelled (5.4))2. Notation. We shall use the standard notation: R to denote thereal numbers; Q to denote the rational numbers; Z to denote the integernumbers; N to denote the natural numbers; and Z+ to denote the nonnegative integers. We use the convenient convention that: R/Z = {x +Z : x ∈ R} (which is homeomorphic to the standard unit circle); R2 /Z2 ={(x1 , x2 ) + Z2 : (x1 , x2 ) ∈ R2 } (which is homeomorphic to the standard 2torus); etc. However, for x ∈ R we denote the corresponding element in R/Zby x (mod 1) (and similarly for R2 /Z2 , etc.).We denote the interior of a subset A of a metric space by int(A), and wedenote its closure by cl(A).If T : X → X denotes a continuous map on a compact metric space thenT n (n ≥ 1) denotes the composition with itself n times.If T : I → I is a C 1 map on the unit interval I = [0, 1] then T 0 denotes itsderivative.3. Prerequisites in point set topology (chapters 1-6).

The first sixchapters consist of various results in topological dynamics for which the onlyprerequisite is a working knowledge of point set topology for metric spaces.For example:Theorem A (Baire). Let X be a compactTmetric space; then if {Un }n∈Nis a countable family of open dense sets then n∈N Un ⊂ X is dense.xixiiPRELIMINARIESTheorem B (sequential compactness). Let X be a metric space;then X is compact if and only if every sequence (xn )n∈N in X contains aconvergent subsequence.Theorem C (Zorn’s lemma). Let Z be a set with a partial ordering.

Ifevery totally ordered chain has a lower bound in Z then there is a minimalelement in Z.Two good references for this material are [4] and [5]4. Pre-requisites in measure theory (chapters 7-12). Chapters 712 form an introduction to ergodic theory, and suppose some familiarity (ifnot expertise) with abstract measure theory and harmonic analysis. Thefollowing results will be required.Theorem D (Riesz representation). There is a bijection between(1) probability measures µ on a compact metric space X (with the Borelsigma algebra),(2) Continuous linear functionals c : C 0 (X) → R,Rgiven by c(f ) = f dµ.Theorem E.

Let (X, B, µ) be a measure space. For every linear functional αR: L1 (X, B, µ) → L1 (X, B, µ) there exists k ∈ L∞ (X, B, µ) such thatα(f ) = f · kdµ, ∀f ∈ L1 (X, B, µ) [3, p.121].In proving invariance of measures in examples the following basic resultwill sometimes be assumed.Theorem f (Kolmogorov extension). Let B be the Borel sigmaalgebra for a compact metric space X. If µ1 and µ2 are two measures for theBorel sigma-algebra which agree on the open sets of X then m1 = m2 [3, p.310].The following terminology will be used in the chapter on ergodic measures.Given two probability measures µ, ν we say that µ is absolutely continuouswith respect to ν if for every set B ∈ B for which ν(B) = 0 we have thatµ(B) = 0.

We write µ << ν and then we have the following result.Theorem G (Radon-Nikodym). If µ is absolutely continuous with re1spect to µ then there exists a (unique)R function f ∈ L (X, B, dν) such thatfor any A ∈ B we can write µ(A) = A f dν.We usually write f = dµdν and call this the Radon-Nikodym derivative of µwith respect to ν.We call two measures µ, ν mutually singular if there exists a set B ∈ Bsuch that µ(A) = 0 and ν(A) = 1.

We then write µ ⊥ ν.In chapter 8 we shall need a passing reference to Lebesgue spaces. ALebesgue space is a measure space which is measurably equivalent to thePRELIMINARIESxiiithe union of unit intervals (with the usual Lebesgue measure) with at mostcountably many points (with non-zero measure).In chapter 11 we shall use the following result.Theorem H (dominated convergence). Let h ∈ L1 (X, B, µ) and let(fn )n∈Z+⊂ L1R(X, B, µ), Rwith |fn (x)| ≤ h(x), converge (almost everywhere) to f (x);then fn dµ → f dµ as n → +∞.Good general references for this material are [1], [2], [3].5.

Subadditive sequences. A simple result which proves its worthseveral times in these notes is the following.Theorem F (subadditive sequences). Let (an )n∈N be a sequence ofreal numbers such that an+m ≤ an + am , ∀n, m ∈ N (i.e. a subadditivesequence); then an → a, as n → +∞, where a = inf{an /n: n ≥ 1}Proof. First note that an ≤ a1 + an−1 ≤ . . . ≤ na1 , and so a ≤ a1 For > 0 we choose N > 0 with aN < N (a + ). For any n ≥ 1 we can writen = kN + r, where k ≥ 0 and 1 ≤ r ≤ N − 1. Thenan ≤ akN + ar ≤ kaN + ar ≤ kaN + sup ar1≤r≤Nand we see thatlim supn→+∞This shows thatkaN + sup1≤r≤N aranaN≤ lim sup=≤ a + .nkNNk→+∞ann→ a, as required.References1.

P. Halmos, Measure Theory, Van Nostrand, Princeton N.J., 1950.2. K. Partasarathy, An Introduction to Probability and Measure Theory, Macmillan, NewDelhi, 1977.3. H. Roydon, Real Analysis, Macmillan, New York, 1968.4. G. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill, New York,1963.5. W.

Sutherland, Introduction to Topological and Metric spaces, Clarendon Press, Oxford, 1975..