Lecture Notes on Dynamical Systems Homeomorphisms on the Circle Diego A. S. Sanhueza October 30, 2018 Abstract In this lecture we lead very quickly the essential results on the Poincar´e'sTheory for maps of the circle. Rotation number is defined and the classification theorem of Poincar´eis proved. Further simples facts also are mentioned. Contents 1 Introduction................................1 2 Preliminaries...............................2 3 Rotation number..............................3 4 The meaning of ρ(f).............................5 4.1 Rational rotation number . .5 4.2 Irrational rotation number . .8 5 Solved problems.............................. 12 x1. INTRODUCTION Homeomorphisms of the circle were first considered by Henri Poincar´ewho used them to obtain qual- itative results for a class of differential equations on the torus. In [15] he introduced the rotation number and in [16] classified those which have a dense orbit by showing that they are topologically conjugated to a rotation by the angle their rotation number. In [4], Arnaud Denjoy shows the existence of orientation- preserving homeomorphisms without periodic orbit and without dense orbit, and the Poincar´e'sresult not assure that are conjugated to a rotation. They are usually called Denjoy C1−diffeomorphisms. Denjoy also proves that such a homeomorphism admits a unique minimal set which is a Cantor set (in contrast to the case when has periodic points. There every minimal set is a periodic orbit). This lead to study of classifying such Cantor sets and in this direction, Nelson Markley [11] characterized those Cantor sets completely in terms of dynamical properties. We mentioned that a Cantor set is always the unique minimal set of some orientation preserving homeomorphism of S1, but it may not be the minimal set of any Denjoy C1−diffeomorphism of S1 (See [14, 17]). Dusa McDuff [12], gives necessary conditions to guarantees when a Cantor set is a minimal set of a Denjoy diffeomorphism, answering partially to an ask posed by Michael Herman [6]. On the other hand, [9] it is showed that complementary intervals of those Cantor sets are permuted freely (without finite cycles) among themselves by the homeomorphism. Continuous map without periodic point were regarded by Joseph Auslander and Y. Katznelson in [2]. If f admits periodic point, the dynamics is very much simple, but in general conjugation to a rotation is not obtained. In [19], M. Zdun gives a necessary and sufficient condition for topological conjugacy of this type of homeomorphisms of the circle. 2 Diego A. S. Sanhueza On the other hand, H. Furstenberg [5] proved a remarkable theorem in connection with ergodic theory. More precisely, he shows that if f : S1 ! S1 is an orientation-preserving homeomorphisms with irrational rotation number and µ is an Borel f−invariant measure, then Fµ ◦ f = Rα ◦ F µ, where α = (Fµ ◦ T )(0) (necessarily irrational). This say that homeomorphisms with irrational rotation number are (metrically) equivalent to an irrational rotation. Moreover, Furstenberg also proved that such homeomorphisms are uniquely ergodic. Diffeomorphisms on the circle give rise to a important object of study within of the theory of dynam- ical systems. We refer to [7, 13] for a more specialized interest. In this short notes, we collect various main results on the Poincar´e'sTheory. The proofs of the principal results are standard and we follow [3, 8] e [18]. x2. PRELIMINARIES Throughout we consider S1 = R=Z. The quotient map is π : R ! R=Z. We denote by π(x) or sometimes x the equivalence class that contains x. We recall that y 2 π(x) if, and only if, x − y 2 Z. It is obvious that each class admits a representant in [0; 1). Definition 2.1. Let f : S1 ! S1 be a continuous function. A lift of f is a continuous function F : R ! R which satisfies (π ◦ F )(x) = (f ◦ π)(x) (1) for all x 2 R. Suppose that F0 is another lift of f, then by (1), we have that (π◦F )(x) = (f ◦π)(x) = (π◦F0)(x), so F (x) − F0(x) = k 2 Z for all x 2 R. By continuity of the lifts, the integral number k = k(F; F0) does not depend on x. In summary, two lift differ by a integer constant, the family of lifts of a given 1 1 continuous function f : S ! S is determined knowing a lift: If F0 is a lift of f, then any other lift is in fF0+k : k 2 Zg. Others important properties of the lifts are summarized in the following proposition. Proposition 2.2. Let f; g : S1 ! S1 be two continuous function and F; G : R1 ! R1 lift of f and g, respectively. Then: (i) F ◦ G is a lift of f ◦ G. In particular, F n is a lift of f n. (ii) Id : R1 ! R1 is a lift of id : S1 ! S1. (iii) If f is a homeomorphism, then F −1 is a lift of f −1. Proof. (i) π[(F ◦ G)(x)] = (π ◦ F )(G(x)) = f(π(G))(x) = [f ◦ g](π(x)) for all x 2 R. (ii) Immediate. (iii) By (i) and (ii) id(π(x)) = π(F ◦ F −1)(x) = f(π(F −1(x)) for all x 2 R. Applying f −1 to this −1 −1 equality, we obtain f (π(x)) = (π ◦ F )(x) for all x 2 R, but this is just the definition of lift. Now, since (π◦F )(x+1) = f(π(x+1)) = f(π(x)) = (π◦F )(x) we see that F (x+1)−F (x) 2 Z and again by continuity of the lift, this difference is independent of x. Moreover, by the preceding paragraph, also is independent of the choice of lift F . We denote this difference by deg(f). Definition 2.3. The number deg(f) is called the degree of f. One can prove the identity: deg(f ◦ g) = deg(f) deg(g) and in particular to obtain deg(f n) = [deg(f)]n for any n ≥ 1. Note also that if f is a homeomorphism, then deg(f) = ±1. Proposition 2.4. The degree of a continuous map is invariant by topological conjugacy. Proof. It is immediate from preceding paragraph, since deg(f) = deg(h−1◦g◦h) = deg(h−1) deg(g) deg(h) −1 and deg(h) = deg(h ) = ±1: Definition 2.5. We say that a homeomorphism f : S1 ! S1 is an orientation-preserving map if it admits a lift which is increasing. 2 Homeomorphisms on the circle 3 Equivalently, f preserves orientation if, and only if, deg(f) = 1. For instance, rotation maps are orientation-preserving homeomorphisms. It is clear that every lift of a map which preserves orientation is increasing. We denote by Homeo+(S1) the set of all the orientation-preserving homeomorphisms on the circle. Proposition 2.6. Let f : S1 ! S1 be an orientation-preserving homeomorphism and F be a lift of f. Then, (i) F (x + k) = F (x) + k for all x 2 R, k 2 Z; (ii) F n(x + k) = F n(x) + k for all x 2 R, k; n 2 Z; n n (iii) If Fk = F + k is another lift, then Fk (x) = F (x) + nk for all x 2 R, k; n 2 Z; n (iv) The map 'n(x) = F (x) − x is periodic and in particular it is bounded (and is independent of the lift). Proof. (i) We note that π(F (x + k)) = f(π(x + k)) = f(π(x)) = π(F (x)), so F (x + k) − F (x) = k0(x) 2 Z for all x 2 R. We shall prove that k = k0(x) for all x 2 R. We suppose first that k > 0 and + 1 prove this by induction. Because f 2 Homeo (S ), the conclusion is clear for k = 1. If k = k0(x) for all x 2 R, then (k + 1)0(x) = F (x + k + 1) − F (x) = F (x + 1) + k − F (x) = k + 1 for all x. This prove that case k > 0. For if k < 0, put k+ = −k and so F (x + k) − F (x) = −(F (x) − F (x − k+)). We now use induction + + in the expression F (x) − F (x − k ) to obtain k0 (x) = −k. This proves (i). (ii) Use induction. (See exercise 2). (iii) Fix k. We first consider n > 0 and shall use induction. For n = 1 there is nothing to prove. n n n+1 n n Suppose Fk (x) = F (x) + nk holds for all x 2 R, then by (ii), Fk (x) = Fk (Fk(x)) = Fk (Fk(x)) + k = F n+1(x) + nk + k = F n+1(x) + (n + 1)k. The case n < 0 is similar. n n (iv) By (ii), 'n(x + 1) = F (x + 1) − (x + 1) = F (x) − x = 'n(x) for all x 2 R: x3. ROTATION NUMBER The most important concept in our study of homeomorphisms of the circle is that of rotation number (Definition 3.3). The following theorem essentially shows their existence and uniqueness. Theorem 3.1. Let f : S1 ! S1 be an orientation-preserving homeomorphism and let F : R ! R be a lift of f. Then, for every x 2 R, the following limit F n(x) − x ρ(F ) = lim (2) n!1 n exists and is independent of x. Proof. We first prove that (2) does not depend on x. Suppose that (2) exists for x 2 R and we write x = [x] + x , where [x] and x denote the integer and fractional part of x, respectively. Then L M L M F n(x) − x F n([x] + x ) − ([x] + x ) lim = lim n!1 n n!1 L Mn L M F n x + [x] − ([x] + x ) = lim n!1 L M n L M F n x − x = lim n!1 L Mn L M Hence (2) also exists for x and they coincide.