Topological Dimension and Dynamical Systems (Universitext)

Topological Dimension and Dynamical Systems (Universitext)

Language: English

Pages: 233

ISBN: 3319197932

Format: PDF / Kindle (mobi) / ePub

Translated from the popular French edition, the goal of the book is to provide a self-contained introduction to mean topological dimension, an invariant of dynamical systems introduced in 1999 by Misha Gromov. The book examines how this invariant was successfully used by Elon Lindenstrauss and Benjamin Weiss to answer a long-standing open question about embeddings of minimal dynamical systems into shifts.

A large number of revisions and additions have been made to the original text. Chapter 5 contains an entirely new section devoted to the Sorgenfrey line. Two chapters have also been added: Chapter 9 on amenable groups and Chapter 10 on mean topological dimension for continuous actions of countable amenable groups. These new chapters contain material that have never before appeared in textbook form. The chapter on amenable groups is based on Følner’s characterization of amenability and may be read independently from the rest of the book.

Although the contents of this book lead directly to several active areas of current research in mathematics and mathematical physics, the prerequisites needed for reading it remain modest; essentially some familiarities with undergraduate point-set topology and, in order to access the final two chapters, some acquaintance with basic notions in group theory. Topological Dimension and Dynamical Systems is intended for graduate students, as well as researchers interested in topology and dynamical systems. Some of the topics treated in the book directly lead to research areas that remain to be explored.

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immediately follows from the observation that if Y is a subset of a topological space X and y ∈ Y then the connected component of y in Y is contained in the connected component of y in X . Proposition 2.5.5 Every product of totally disconnected spaces is itself totally disconnected. Proof Let (X i )i∈I be a family of totally disconnected spaces and consider their direct product X := i∈I X i . Let C be a non-empty connected subset of X . As the continuous image of a connected space is itself

is bounded implies that σ(t) ∈ / for |t| large enough. By applying Lemma 5.1.2 with A = σ −1 ( ) and B = σ −1 (X \ ), we deduce that we can find a, b ∈ Q such that σ(a) ∈ , σ(b) ∈ X \ and d(σ(a), σ(b)) = |a − b| ≤ 1/(i + 1). Consequently, we can take ri+1 = a. This completes our induction. Consider now the sequence r = (rn )n≥1 . We have that r ∈ X . Indeed, since is bounded, there exists a constant M ≥ 0 such that u ( j) ≤ M for all j ≥ 1. This implies r12 + r22 + · · · + r 2j ≤ M 2 for all j ≥

of block-type associated with (q, B). Show that the set of periodic points of the dynamical system (X, σ) is dense in X . 8.4 (Adding machines). Let (an )n∈N be a sequence of positive integers. Consider the product space {0, 1, . . . , an − 1} X := n∈N and the map T : X → X defined in the following way. If x = (xn )n∈N ∈ X with xn = an − 1 for all n ∈ N then we take T (x) := (yn )n∈N , where yn = 0 for all n ∈ N. Otherwise, there is a largest integer n 0 ∈ N such that xn = an −1 for all n ≤ n 0

10.2 Definition of Mean Topological Dimension 193 Given a finite open cover α = (Ui )i∈I of X and an element A ∈ P f in (G), we define the finite open cover α A = α A (X, G, T ) by Tg−1 (α). α A := (10.2.1) g∈A Formally, α A is the family indexed by I A (the set consisting of all maps from A to I ) formed by all the open subsets Tg−1 (Uι(g) ) ⊂ X, g∈A where ι : A → I runs over I A . Proposition 10.2.1 Let X be a topological space equipped with a continuous action T : G × X → X of a group

G. Let α be a finite open cover of X . Then the following hold: (i) D(α Ag ) = D(α A ) for all g ∈ G and A ∈ P f in (G); (ii) D(α A ) ≤ D(α B ) for all A, B ∈ P f in (G) such that A ⊂ B; (iii) if X is normal then one has D(α A∪B ) ≤ D(α A ) + D(α B ) for all A, B ∈ P f in (G). Proof If A ∈ P f in (G) and g ∈ G, then −1 Thg (α) α Ag = h∈A (Th ◦ Tg )−1 (α) = h∈A Tg−1 ◦ Th−1 (α) = h∈A Tg−1 (Th−1 (α)) = h∈A Th−1 (α) = Tg−1 (by Proposition 6.1.1) h∈A = Tg−1 (α A ). This shows that the

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