Fixed Points of Functions Analytic in the Unit Disk Carl C. Cowen IUPUI (Indiana University Purdue University Indianapolis) IUPUI Grad Student Seminar, October 28, 2011 Fixed Points of Functions Analytic in the Unit Disk Carl C. Cowen IUPUI (Indiana University Purdue University Indianapolis) IUPUI Grad Student Seminar, October 28, 2011 Some of this is joint work with Christian Pommerenke (1982). Definition (Preliminary) If f is a function mapping some set into itself, a point x0 in the set is called a fixed point of f if f(x0) = x0. Two well known fixed point theorems are: Theorem (Brouwer) If f is a continuous function mapping the closed unit ball B in RN into itself, then f has a fixed point in B. Contraction Mapping Theorem Suppose X is a compact metric space. If g is a continuous function mapping X into itself and satisfies jg(x) − g(y)j < jx − yj for all x and y in X with x =6 y, then g has a unique fixed point in X. Today, I want to consider the fixed points an analytic function, ', that maps the unit disk D into itself. The Schwarz-Pick Lemma implies ' has at most one fixed point in D and some maps, like 1 1 '(z) = z + 2 2 have none in the open disk! Of course, if ' is a continuous map of D into D, then ' must have a fixed point in D. We only assume ' is analytic on D, the open disk! and it is possible that ' cannot be extended to be continuous at any point of the unit circle! On the other hand, we can try to define a function on the unit circle, by '(eiθ) = lim '(reiθ) b r!1− Perhaps surprisingly, if ' is a bounded map on D, and in particular, maps of the disk into itself, the radial limit function 'b is defined almost everywhere in the sense of Lebesgue measure on the circle. (Because it rarely leads to misunderstandings, we abuse the notation and write iθ iθ '(e ) for 'b(e ). ) This leads us to an extension of the definition of fixed point: Definition(Extended) Suppose ' is an analytic map of D into itself. If jbj < 1, we say b is a fixed point of ' if '(b) = b. If jbj = 1, we say b is a fixed point of ' if limr!1− '(rb) = b. Julia-Caratheordory Theorem implies 0 If b is a fixed point of ' with jbj = 1, then limr!1− ' (rb) exists (call it '0(b)) and 0 < '0(b) ≤ 1. Denjoy-Wolff Theorem (1926) If ' is an analytic map of D into itself, not the identity map, there is a unique fixed point, a, of ' in D such that j'0(a)j ≤ 1. This distinguished fixed point will be called the Denjoy-Wolff point of '. The Schwarz-Pick Lemma implies ' has at most one fixed point in D and if ' has a fixed point in D, it must be the Denjoy-Wolff point. Examples (1) '(z) = (z + 1=2)=(1 + z=2) is an automorphism of D fixing 1 and −1. The Denjoy-Wolff point is a = 1 because '0(1) = 1=3 (and '0(−1) = 3) (2) '(z) = z=(2 − z2) maps D into itself and fixes 0, 1, and −1. The Denjoy-Wolff point is a = 0; '0(0) = 1=2 (and '0(±1) = 3) (3) '(z) = (2z3 + 1)=(2 + z3) is an inner function fixing fixing 1 and −1 with Denjoy-Wolff point a = 1 because '0(1) = 1 (and '0(−1) = 9) (4) Inner function '(z) = exp(z + 1)=(z − 1) has a fixed point in D, Denjoy-Wolff point a ≈ :21365, and infinitely many fixed points on @D Today, discuss questions about fixed points of analytic maps of D into D and the values of the derivative at the fixed points. We have examples of maps with several fixed points in the closed disk. Schwarz-Pick and Denjoy-Wolff say that all but one of these fixed points satisfies '0(b) > 1. Questions about ', analytic and '(D) ⊂ D: • How many fixed points can ' have? with ' univalent? • How small can '0(b) be? with ' univalent? Definition: If ' is an analytic map of the unit disk into itself, the fixed point set of ' is the set F = z 2 D : lim '(rz) = z r!1− Theorem: If ' is an analytic function that maps the unit disk into itself, then the part of the fixed point set of ' in @D has linear measure zero. Example: Let K be a compact set of measure zero in @D. There is a function ' analytic in D and continuous on D such that '(D) ⊂ D and the fixed point set of ' is f0g [ K. Theorem: If ' is a univalent analytic function that maps the unit disk into itself, then the fixed point set of ' has capacity zero. Example: Let K be a compact set of capacity zero in @D and a be a point of @D n K. There is a function ' analytic and univalent in D such that '(D) ⊂ D and the fixed point set of ' is fag [ K. In the Theorem below, we assume that the Denjoy-Wolff point, a = b0 has been normalized so that b0 = 0 or b0 = 1. Theorem: Let ' be an analytic function with '(D) ⊂ D and suppose b0; b1; ··· ; bn are fixed points of '. (i) If b0 = 0 then n X 1 1 + '0(0) ≤ Re '0(b ) − 1 1 − '0(0) j=1 j Theorem (cont'd): 0 (ii) If b0 = 1 and 0 < ' (1) < 1 then n X 1 '0(1) ≤ '0(b ) − 1 1 − '0(1) j=1 j 0 (iii) If b0 = 1 and ' (1) = 1 then n 2 X j1 − bjj 1 ≤ 2 Re − 1 '0(b ) − 1 '(0) j=1 j Moreover, equality holds if and only if ' is a Blaschke product of order n + 1 in case (i) or order n in cases (ii) and (iii). Note that, if ' has infinitely many fixed points, then the appropriate inequality holds for any choice of finitely many fixed points. In particular, only countably many of the fixed points of ' can have finite angular derivative. 0 If b0; b1; b2; ··· are the countably many fixed points for which ' (bj) < 1, then the corresponding infinite sum converges and the appropriate inequality holds. About the proof: First, notice that '0(b) = 1 at a fixed point b is a fixed point with \multiplicity greater than 1" in the sense that b fixed means b is a root of '(z) − z = 0 and '0(b) − 1 = 0 means it is a root of '(z) − z = 0 with multiplicity greater than 1. The proof is based on the following fact: If r(z) is a rational function of order n with distinct fixed points p1; p2; ··· ; pn; pn+1 in the sphere, then n+1 X 1 = 1 1 − r0(p ) j=1 j The inequalities for univalent maps seem related to iteration of the maps. Five different types of maps of D into itself from the perspective of iteration, classified by the behavior of the map near the Denjoy-Wolff point, a In one of these types, '0(a) = 0, (e.g., '(z) = (z2 + z3)=2 with a = 0), a model for iteration doesn't seem useful for studying these inequalities. In the other four types, when '0(a) =6 0, the map ' can be intertwined with a linear fractional map and classified by the possible type of intertwining: σ intertwines Φ and ' in the equality Φ ◦ σ = σ ◦ ' We want to do this with Φ linear fractional and σ univalent near a, so that σ is, locally, a change of variables. Using the notion of fundamental set, this linear fractional model becomes essentially unique [Cowen, 1981] Φ σ σ ϕ A linear fractional model in which ' maps D into itself with a = 1 and 1 1 '0(1) = , σ maps into the right half plane, and Φ(w) = w 2 D 2 Linear Fractional Models: • ' maps D into itself with '0(a) =6 0 (' not an elliptic automorphism) • Φ is a linear fractional automorphism of Ω onto itself • σ is a map of D into Ω with Φ ◦ σ = σ ◦ ' I. (plane dilation) jaj < 1, Ω = C, σ(a) = 0, Φ(w) = '0(a)w II. (half-plane dilation) jaj = 1 with '0(a) < 1, Ω = fw : Re w > 0g, σ(a) = 0, Φ(w) = '0(a)w III. (plane translation) jaj = 1 with '0(a) = 1, Ω = C, Φ(w) = w + 1 IV. (half-plane translation) jaj = 1 with '0(a) = 1, Ω = fw : Im w > 0g, (or Ω = fw : Im w < 0g), Φ(w) = w + 1 Linear Fractional Models: • ' maps D into itself with '0(a) =6 0 (' not an elliptic automorphism) • Φ is a linear fractional automorphism of Ω onto itself • σ is a map of D into Ω with Φ ◦ σ = σ ◦ ' I. (plane dilation) jaj < 1, Ω = C, σ(a) = 0, Φ(w) = '0(a)w II. (half-plane dilation) jaj = 1 with '0(a) < 1, Ω = fw : Re w > 0g, σ(a) = 0, Φ(w) = '0(a)w (equivalent to strip translation) III. (plane translation) jaj = 1 with '0(a) = 1, Ω = C, Φ(w) = w + 1 f'k(0)g is NOT an interpolating sequence (i.e. f'k(0)g close together) IV. (half-plane translation) jaj = 1 with '0(a) = 1, Ω = fw : Im w > 0g, (or Ω = fw : Im w < 0g), Φ(w) = w + 1 f'k(0)g IS an interpolating sequence (i.e. f'k(0)g far apart) The constructions involved give σ is univalent if and only if ' is univalent.