Dynamics of unimodal interval maps Notes for the mini-course at the University of Vienna, May 2017 Ana Anuˇsi´c,University of Zagreb Version of May 4, 2017 1 Discrete dynamical systems - basic definitions Definition 1.1. Dynamical system is a pair (X; f), where X is a compact metric space and f : X ! X is (piecewise) continuous. For n 2 N denote by f n := f ◦f ◦:::◦f (n times). The forward orbit of x 2 X is Orb(x) := fx; f(x); f 2(x); f 3(x);:::g: The !-limit set of x is a set of accumulation points of Orb(x), i.e., ni !(x; f) = fy 2 X : there exists a strictly increasing (ni)i2N; f (x) ! y as i ! 1g: We say that x is periodic if there exists n 2 N such that f n(x) = x. The smallest such n 2 N is called a prime period of x. If x has period one, it is called a fixed point. If x is not periodic but there exists m 2 N such that f m(x) is periodic, then x is called preperiodic. If x 2 !(x), then x is called recurrent. Definition 1.2. Let (X; f) and (Y; g) be dynamical systems. We say that the systems (or maps f, g) are topologically conjugate if there exists a homeomorphism h: X ! Y such that h ◦ f = g ◦ h. Such h is called a topological conjugation of f and g. If h can be taken at most continuous and surjective, systems are topologically semi-conjugate and g is called a factor of f. See Figure 1. Remark 1.3. Conjugation h maps orbits of X to orbits of Y . Also h(!(x; f)) = !(h(x); g) for all x 2 X. Later we will see more invariants under topological conjuga- tion. Often one thinks of conjugate systems as dynamically equivalent. 1 f X X h h g Y Y Figure 1: Topological (semi)-conjugation of f and g. 2 Unimodal maps We will be studying dynamical properties of systems (I; f), where I = [0; 1] and f : I ! I is unimodal, defined below. Definition 2.1. We say that f : I ! I is unimodal if (a) f is continuous, (b) there exists a unique local maximum c 2 (0; 1), i.e., fj[0;c) is strictly increasing, fj(c;1] is strictly decreasing, (c) f(0) = f(1) = 0. Example 2.1. Typical families of unimodal maps are (a) The logistic family fa(x) = ax(1 − x), a 2 [0; 4]. (b) The tent family Ts(x) = minfsx; s(1 − x)g, s 2 [0; 2]. (c) The sine family Sα(x) = α sin(πx), α 2 [0; 1]. See Figure 2. Note that fa;Ts;Sα have a fixed point 0 for all parameters. Also, since a s 4 2 α c c c (a) (b) (c) Figure 2: Graphs of (a) f3,(b) T1:5,(c) S0:75. f4(1=2) = T2(1=2) = S1(1=2) = 1, and f4(1) = T2(1) = S1(1) = 0, c = 1=2 is prefixed for f4;T2;S1. For graphical representation of orbits see a cobweb diagram in Figure 3. 2 s 2 x c Figure 3: Cobweb diagram gives a graphical representation of an orbit. Take x 2 I and move up until you hit the graph of f. Then move horizontally until you hit the diagonal. That point if (f(x); f(x)). Continue vertically until you hit the graph and again horizontally until you hit the diagonal and obtain a point (f 2(x); f 2(x)). What is !(x)? Remark 2.2. Maps f4;T2 and S1 are all topologically conjugate. Conjugation between πx 2 f4 and T2 is given by h(x) = sin( 2 ) . Note that e.g. f2 and T1 are not conjugate since T1 has an interval of fixed points while f2 has two. In general there exist many parameters a for which fa is conjugate to some tent map, but the conjugacies will not be smooth as in a = 4 case. As we will see later, every unimodal map is semi-conjugate to some tent map. 3 Period doubling route to chaos Definition 3.1. Let x0 be a periodic point of f with prime period n. We say that x0 is d n (a) attracting if j dx f (x0)j < 1, d n (b) neutral if j dx f (x0)j = 1, d n (c) repelling if j dx f (x0)j > 1. Remark 3.2. By the Mean Value Theorem, if x0 is attracting, there exists an open nk set U 3 x0 such that limk!1 f (x) = x0 for all x 2 U. If x0 is repelling, there exists kn U 3 x0 open such that for every x0 6= x 2 U there exists k 2 N such that f (x) 62 U. Example 3.1. (Period doubling cascade in the logistic family) For a < 1 =: a1 point 0 is attracting fixed point of fa. Fixed point 0 is repelling for all a > 1 and neutral when a = 1. If 1 < a < 3 := a2 then fa has an attracting fixed point xa which attracts all x 2 (0; 1). 3 That point becomes neutral when a = 3 and repelling when a > 3. For 3 < a < 3:449489 ::: =: a4 there is an attracting period 2 cycle of fa which becomes neutral in a = a4 and repelling when a > a4, creating a new attracting period 4 cycle. n Further calculations show that the attracting 2 -cycle becomes neutral at a2n+1 and n+1 repelling when a > a2n+1 , creating an attracting period 2 cycle. Parameters are ap- proximately a8 = 3:54409 :::, a16 = 3:56440726 :::, a32 = 3:56875 :::, a64 = 3:56969 :::, . The limit limn!1 a2n =: afeig ≈ 3:569945672 ::: is called a Feigenbaum param- eter. Original results appeared in [14], [11]. Remark 3.3. A qualitative change in the behavior of the system as in e.g. a = 1, where a single neutral periodic orbit appears and splits into stable and unstable periodic orbit is called a saddle-node bifurcation. A change as in e.g. an, n ≥ 2, where a single attracting periodic orbit breaks into an attractive period 2 cycle is called a period- doubling bifurcation. 2n 2n+1 i Note that for a < a < a a map fa has a single periodic cycle of prime period 2 n n for every i = 0;:::; 2 and the 2 -cycle is attracting. Map fafeig has periodic orbits of n prime period 2 for all n 2 N0 and an attracting Cantor set (as we will later see). See Figure 4. Figure 4: Period doubling in logistic family. Picture is taken from [17]. an+1−an Numerics also indicate that limn!1 = δ = 4:669201609 :::. What is fasci- an+2−an+1 nating is that δ is universal for families of unimodal maps which are smooth enough. Usually the smoothness assumption is the negativity of Schwarzian derivative, see the definition below. Constant δ is called the (first) Feigenbaum constant. The univer- sality of δ was first noticed numerically (see [14],[11]) and explained using the renor- malization theory, see e.g. [23] and the next section. 4 Definition 3.4. Let f : I ! I be continuous and at least three times differentiable. The Schwarzian derivative of f at x is f 000(x) 3 f 00(x)2 Sf(x) = − : f 0(x) 2 f 0(x) If f 0(x) = 0 we define Sf(x) = −∞. Unimodal map f for which Sf(x) < 0 for all x 2 I is called S-unimodal. Remark 3.5. If Sf < 0, then f 0 cannot have a positive local minimum or negative local maximum. So between two successive extrema of f 0 there must be a critical point of f. Example 3.2. Every map in logistic family and sine family is S-unimodal. Tent map is not S-unimodal since it is not smooth at the critical point. Theorem 3.6 (Singer, [30]). If S-unimodal map has an attracting period orbit, then it attracts the critical point c. Thus, S-unimodal map has at most one attracting periodic orbit. Remark 3.7. More generally, attracting orbits of interval maps with negative Schwarzian derivative attract a critical point or a boundary point. So, if Sf < 0 and f has n critical points, then the number of attracting periodic orbits is at most n + 2. Note that the definition of unimodal maps requires f(0) = f(1) = 0. 4 Renormalization Definition 4.1. Unimodal map f is called renormalizable if there exists a closed interval J ⊂ I and n ≥ 2 such that (i) f n(J) ⊂ J (ii) J, f(J),... f n−1(J) have disjoint interiors (iii) J contains c in its interior. n Interval J is called a restrictive interval of period n and f jJ : J ! J is called the return map or renormalization of f to J. n Remark 4.2. Note that f jJ is again unimodal (possibly turned 'upside down', i.e., c is n −1 the minimum). Denote by ': J ! I an affine surjection such that '◦f jJ ◦' : I ! I n −1 is unimodal (c is the maximum). Then f 7! R(f; J) = ' ◦ f jJ ◦ ' is called a renormalization operator. Note that R(f; J) can again be renormalizable. In that case we say that f is twice renormalizable and analogously n-times renormalizable (n = 1 is also allowed). See Figure 5.