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INTERVAL EXCHANGE TRANSFORMATIONS Contents 1. Rotations and Tori 2 1.1. Rotation and Their Coding 2 1.2. Induced Map, Substituti

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INTERVAL EXCHANGE TRANSFORMATIONS Contents 1. Rotations and Tori 2 1.1. Rotation and Their Coding 2 1.2. Induced Map, Substituti INTERVAL EXCHANGE TRANSFORMATIONS VINCENT DELECROIX Abstract. These are lecture notes for 4 introductory talks about interval exchange transformations and translation surfaces given by the author in Salta (Argentina) in November 2016. Contents 1. Rotations and tori 2 1.1. Rotation and their coding 2 1.2. Induced map, substitutions and continued fractions 4 1.3. Suspensions 5 1.4. Further results 6 2. Interval exchange transformations and translation surfaces 7 2.1. Interval exchange transformations 8 2.2. Rauzy induction 8 2.3. Keane theorem 9 2.4. Translation surfaces and suspensions of iet 10 2.5. Strata, Teichm¨ullerflow and Rauzy-Veech induction 11 2.6. Best approximations 12 2.7. Notes and further results 12 3. Equidistribution 12 3.1. Crash course in ergodic theory 13 3.2. Invariant measures of interval exchange transformations 14 3.3. Linear recurrence and Boshernitzan condition 15 3.4. Vorobets identities 16 3.5. Notes and further results 19 4. Some generic properties of interval exchange transformations 19 4.1. How do we prove something for a generic translation surface? 19 4.2. Some results that hold for all translation surfaces 19 4.3. Masur asymptotic theorem for tori 20 4.4. Sketch of a proof of Kerckhoff-Masur-Smillie theorem 20 4.5. Notes and further results 21 5. Further reading and some open questions 21 6. Exercises 21 6.1. Word combinatorics and coding of interval exchange transformations 21 6.2. Permutations and Rauzy diagrams 22 6.3. Dynamics 23 6.4. Rotations 23 6.5. Linear recurrence and Boshernitzan condition 24 References 25 In this course, we will see interval exchange transformations from different perspectives, namely: (1) as a map of the interval, (2) as a Poincar´emap of a flow on a surface, (3) as a symbolic dynamical system. 1 2 VINCENT DELECROIX The aim of this course is to give an understanding of the interplay between these different point of views. In the first course we will only consider rotations (or Sturmian languages) and will explore the link with continued fractions and SL(2; R)= SL(2; Z). In the second course, we introduce the main actors: interval exchange transformations, translation surfaces, Rauzy induction and the SL(2; R)-action. We will give a proof of Keane minimality condition. This second course can be seen as generalization of what was done in the first one. In the third lecture we will introduce invariant measures and related concepts (linear recurrence and Boshernitzan condition). In the last lecture we discuss three deep results: Masur's asymptotic about counting, Kerckhoff-Masur-Smille theorem about generic unique ergodicity and a theorem about linear recurrence due to Kleinbock-Weiss and then improved by Chaika-Cheung-Masur. 1. Rotations and tori 1.1. Rotation and their coding. Let α (0; 1) and let us consider the following map of the unit interval T : x x + α(mod 1). In other words 2 α 7! x + α if x < 1 α T (x) = x + α 1 if x > 1 − α : α 8 undefined− if x = 1 − α < − : A B x0 x3 x6 x1 x4 x7 x2 x5 coding: u = AABAABAB... B A Figure 1. A picture of the rotation by α = (3 p5)=2 and the orbit of x0 = 0 (here xn n − denotes T (x0)). In dynamics, we are interested in the behavior of orbits under iteration. Namely, given an initial condition x [0; 1] how does look like the sequence T (x), T 2(x), T 3(x), . ? Is it dense? Is it equidistributed? 2 One way to proceed, is to introduce a coding. The map Tα naturally induces a partition of the unit interval top top in two subintervals IA = [0; 1 α) and IB = (1 α; 1]. Given an initial condition x [0; 1] we associates its coding which is the sequence− u = u u ::: on −A; B defined by 2 0 1 f g A if T n(x) I u = A : n B if T n(x) 2 I 2 B As an example, the coding of the orbit x = 0 under T with α = (3 p5)=2 is α − u = AABAABABAABAABABAABABAABAABABAABAA : : : : The natural coding or language of the map Tα is the set of finite words that appear in some coding. One can show that for the rotation of Figure 1 one has L = "; A; B; AA; AB; BA; AAB; ABA; BAA; BAB; : : : . α f g Given Lα and a non-negative integer n we denote by Lα,n the set of words of length n in Lα. Given a word u = u0u1 : : : un 1 Lα we can associate the set of points in I whose orbit start with u, namely − 2 top top 1 top 2 top (n 1) top Iu = I0 T − (Iu1 ) T − (Iu2 ) ::: T − − (Iun 1 ): \ \ \ \ − For each n, the sets (Iu)u Lα,n form a partition of the interval (up to the extremity of these intervals). 2 1 bot bot We can also define a partition for T − given by IA = [0; α) and IB = (α; 1] and similarly, for u = u0u1 : : : un 1 Lα the following interval − 2 bot bot bot n 1 bot Iu = Iun 1 T (Iun 2 ) ::: T − (Iu0 ): − \ − \ \ n top bot By construction, T maps by translation Iu to Iu (see Figure 2). More generally, a language L is a non-empty set of words on a finite set called alphabet that: is factorial: if u = u0u1 : : : un 1 belongs to L then u1 : : : un 1 and u0 : : : un 2 belongs to L, • is prolongable: for all u L −there exists a letter a such that− au L and a− letter b so that ub L. • 2 2 2 INTERVAL EXCHANGE TRANSFORMATIONS 3 AB BA BB ABA ABB BAB BBA ABAB ABBA BABA BABB BBAB ABABB ABBAB BABAB BABBA BBABA BBABB BB AB BA ABB BAB BBA ABA BBAB BABB ABAB ABBA BABA ABBAB BBABB ABABB BABAB BABBA BBABA ABABBA ABBABA ABBABB BABABB BABBAB BBABAB BBABBA ABABBAB ABBABAB ABBABBA BABABBA BABBABA BABBABB BBABABB BBABBAB ABBABB BABABB BABBAB BBABAB BBABBA ABABBA ABBABA BABBABB BBABABB ABABBAB ABBABAB ABBABBA BABABBA BABBABA BBABBAB Figure 2. A picture for n = 2; 3; 4; 5; 6; 7 and α = (3 p5)=2 of the partitions induced by top bot n − (Iu )u Lα,n and (Iu )u Lα,n . The map T is a translation reduced to each of the subinterval 2 2 Itop by either nα or 1 nα . u f g − f g The complexity function of a language L is the function pL which to a non-negative integer associates the number of words of length n in L. A language is called uniformly recurrent if for all positive integer n there exists an N so that any word of length N in L contains all words of length n as factors. This property is equivalent to the minimality (or density of orbits) of the underlying dynamical system (see Exercise 2). A language L is said to be k-balanced 1 if for any pair of words u; v L of the same length and any 2 letter α we have u α v α k. This property is related to invariant measures that will be discussed in Section 3. jj j − j j j ≤ Theorem 1. Let Lα be the language of a rotation by an irrational number α. Then Lα (1) has complexity function p(n) = n + 1, (2) is 1-balanced, (3) is uniformly recurrent (in other words, all infinite orbits of Tα are dense). n Proof. The words of length n are exactly the number of intervals that Tα is made of. The limit points of k these intervals are exactly 0; 1 and the preimages T − (1 α) for k = 0; 1; : : : ; n. As α is irrational, these preimages are all different and we hence obtain n + 2 different− points that define n + 1 intervals. n By definition Tα (x) = x + nα where x = x x is the fractional part of x. It is easily seen that the coding of x is given by f g f g − b c A if x + (n + 1)α x + nα = 0, u = n B if bx + (n + 1)αc − bx + nαc = 1. b c − b c Hence, for the coding u of x we have nα if x > 1 nα u0u1 : : : un 1 B = x + nα = b c − f g : j − j b c nα + 1 if x < 1 nα . b c − f g Hence the language is 1-balanced. n Uniform recurrence of the language Lα is equivalent to the fact that all infinite orbits of T (α) are dense (see Exercise 2). We can always build a sequence of integers q so that q α 0 (one can use Dirichlet n ! 1 f n g ! (or pigeonhole) principle). It follows that the sequence mqnα n 0;1 m<1= qnα is dense. So is the orbit of 0. Now to prove that every orbit is dense, it is enough tof remarkg ≥ that≤T n(x)f = gx + nα = T n(0) + x . f g f g 1Sometimes, author uses balanced for 1-balanced. 4 VINCENT DELECROIX 1.2. Induced map, substitutions and continued fractions. Given a dynamical system T : X X and n ! a subset Y X we can define the return time: r(x) = rY (x) = inf n > 0 : T (x) Y . If it is well defined in Y we can⊂ define an induced map T by setting for f 2 g jY T (x) = T r(x)(x); for x Y . jY 2 This is a very important notion to study dynamical system in general.
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