Within each chapter results Theorems, Propositions or Lemmas are la- belled by the chapter and then the order of occurrence e.g.. sublem-We denote the end of a proof by.. Finally, equati
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Trang 71 Conventions The book is divided into 16 chapters, each subdivided into sections numbered in order (e.g chapter 12, section 3 is numbered 12.3) Within each chapter results (Theorems, Propositions or Lemmas) are la- belled by the chapter and then the order of occurrence (e.g the fifth result
in chapter 3 is Proposition 3.5) The exceptions to this rule are: mas which are presented within the context of the proof of a more important result (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).
sublem-We denote the end of a proof by .
Finally, equations are numbered by the chapter and their order of rence (e.g the fourth equation in chapter 5 is labelled (5.4))
occur-2 Notation We shall use the standard notation: R to denote the real numbers; Q to denote the rational numbers; Z to denote the integer numbers; N to denote the natural numbers; and Z + to denote the non- negative integers We use the convenient convention that: R/Z = {x +
Z : x ∈ R} (which is homeomorphic to the standard unit circle); R 2 /Z 2 = {(x 1 , x 2 ) + Z 2 : (x 1 , x 2 ) ∈ R 2
} (which is homeomorphic to the standard torus); etc However, for x ∈ R we denote the corresponding element in R/Z
2-by x (mod 1) (and similarly for R 2 /Z 2 , etc.).
We denote the interior of a subset A of a metric space by int(A), and we denote its closure by cl(A).
If T : X → X denotes a continuous map on a compact metric space then
T 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 its derivative.
3 Prerequisites in point set topology (chapters 1-6) The first six chapters consist of various results in topological dynamics for which the only prerequisite is a working knowledge of point set topology for metric spaces For example:
Theorem A (Baire) Let X be a compact metric space; then if {U n } n ∈N
is a countable family of open dense sets then T
n ∈N U n ⊂ X is dense.
xi
Trang 8xii PRELIMINARIES
Theorem B (sequential compactness) Let X be a metric space; then X is compact if and only if every sequence (x n ) n ∈N in X contains a convergent subsequence.
Theorem C (Zorn’s lemma) Let Z be a set with a partial ordering If every totally ordered chain has a lower bound in Z then there is a minimal element in Z.
Two good references for this material are [4] and [5]
4 Pre-requisites in measure theory (chapters 12) Chapters
7-12 form an introduction to ergodic theory, and suppose some familiarity (if not expertise) with abstract measure theory and harmonic analysis The following 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 Borel sigma algebra),
(2) Continuous linear functionals c : C 0 (X) → R,
given by c(f ) = R f dµ.
Theorem E Let (X, B, µ) be a measure space For every linear tional α : L 1 (X, B, µ) → L 1 (X, B, µ) there exists k ∈ L ∞ (X, B, µ) such that α(f ) = R f · kdµ, ∀f ∈ L 1 (X, B, µ) [3, p.121].
func-In proving invariance of measures in examples the following basic result will sometimes be assumed.
Theorem f (Kolmogorov extension) Let B be the Borel algebra for a compact metric space X If µ 1 and µ 2 are two measures for the Borel sigma-algebra which agree on the open sets of X then m 1 = m 2 [3, p 310].
sigma-The following terminology will be used in the chapter on ergodic measures Given two probability measures µ, ν we say that µ is absolutely continuous with respect to ν if for every set B ∈ B for which ν(B) = 0 we have that µ(B) = 0 We write µ << ν and then we have the following result.
Theorem G (Radon-Nikodym) If µ is absolutely continuous with spect to µ then there exists a (unique) function f ∈ L1(X, B, dν) such that for any A ∈ B we can write µ(A) = R
Trang 9PRELIMINARIES xiii
the union of unit intervals (with the usual Lebesgue measure) with at most countably many points (with non-zero measure).
In chapter 11 we shall use the following result.
Theorem H (dominated convergence) Let h ∈ L 1 (X, B, µ) and let (f n ) n ∈Z +
⊂ L 1 (X, B, µ), with |f n (x) | ≤ h(x), converge (almost everywhere) to f(x); then R f n dµ → R f dµ as n → +∞.
Good general references for this material are [1], [2], [3].
5 Subadditive sequences A simple result which proves its worth several times in these notes is the following.
Theorem F (subadditive sequences) Let (a n ) n ∈N be a sequence of real numbers such that a n+m ≤ a n + a m , ∀n, m ∈ N (i.e a subadditive sequence); then a n → a, as n → +∞, where a = inf{a n /n: n ≥ 1}
Proof First note that a n ≤ a 1 + a n −1 ≤ ≤ na 1 , and so a ≤ a 1 For
> 0 we choose N > 0 with a N < N (a + ) For any n ≥ 1 we can write
n = kN + r, where k ≥ 0 and 1 ≤ r ≤ N − 1 Then
1 P Halmos, Measure Theory, Van Nostrand, Princeton N.J., 1950.
2 K Partasarathy, An Introduction to Probability and Measure Theory, Macmillan, New Delhi, 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, ford, 1975.
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4 Pre-requisites in measure theory (chapters 12) Chapters
7-12 form an introduction to ergodic theory, and suppose some familiarity... Then
1 P Halmos, Measure Theory, Van Nostrand, Princeton N.J., 1950.
2 K Partasarathy, An Introduction to Probability and Measure Theory, Macmillan, New Delhi, 1977....
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, ford,