LECTURE 2. defined recursively by x i+1 := f λ (x i ) with starting point x 0 = 1/2. If we plot the set of accumulation points of P λ, that is,

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1 LECTURE 2 1. Rational maps Last time, we considered the dynamical system obtained by iterating the map x f λ λx(1 x). We were mainly interested in cases where the orbit of the critical point was periodic. In general, the post-critical set is the sequence P λ := {x i i Z + } defined recursively by x i+1 := f λ (x i ) with starting point x 0 = 1/2. If we plot the set of accumulation points of P λ, that is, Q λ := P λ = {x R ε > 0, (x ε, x) (x, x + ε) P λ } against λ, we obtain what is called a bifurcation diagram. Figure 1. Bifurcation diagram for the logistical map Initially, the orbit of x 0 attracted to a fixed point of f λ. As λ grows, an attracting 2-cycle appears, and then succesive period doublings occur. One way to generalize this is to let λ and x be complex. It is conventional to instead look at the polynomial g c : C C z z 2 + c. The critical point of g c is z 0 = 0, and as before, we are interested in the sequence (z i ) i=0 recursively defined by z i+1 := g c (z i ). The so-called Mandelbrot set M is the set of c C for which the sequence (z i ) i=0 stays bounded. More generally, we can look at iterations of arbitrary complex polynomials, or even rational maps, that is, quotients of polynomials. One way to understand the large space of dynamical systems generated by rational maps is to find combinatorial 1

2 2 LECTURE 2 Figure 2. The Mandelbrot set descriptions of them, like the permutation σ associated to a critically periodic f λ. Then we hope to have associated existence and uniqueness statements. Suppose f : Ĉ Ĉ is holomorphic (and thus a rational map). Its critical set is and its post-critical set is C f := {z Ĉ D zf : T z Ĉ T f(z) Ĉ is the 0 map} P f := {f n (z) n Z +, z C f } = We say that f is postcritically finite (PF) if P f <. f n (C f ). n=1 Note that f(p f ) P f and in general the inclusion can be strict. 2. Julia sets and Hubbard trees If f : C C is holomorphic and proper (and thus a complex polynomial) then we define its filled Julia set as K f := {z C the sequence ( f n (z) ) n=1 is bounded}. Suppose that P f <. Then P f K f. Moreover, K f is compact, connected, and has connected complement. This implies that K f is simply connected and therefore between any two points of P f there is a unique homotopy class of arcs connecting them through K f. We thus strip the filled Julia set K f down to a tree T f connecting the points in P f, called the Hubbard tree. Then the induced map f : T f T f gives simple combinatorial data from which it is possible to recover f, just as we could recover f λ from the permutation σ in the first lecture. 3. Thurston s rigidity Suppose f, g : Ĉ Ĉ are postcritically finite rational maps. We say that f and g are Thurston equivalent or homotopy conjugate if there exist orientation-preserving homeomorphisms h, h : (Ĉ, P f ) (Ĉ, P g)

3 LECTURE 2 3 Figure 3. A filled Julia set Figure 4. The associated Hubbard tree 3 such that h is homotopic to h rel P f and h f = g h. Moreover, we say that h is a Thurston equivalence. Theorem 3.1. Suppose that f and g are postcritically finite rational maps and h : (Ĉ, P f ) (Ĉ, P g) is a Thurston equivalence. Then either there exists a unique conformal homeomorphism (Möbius transformation) ĥ : (Ĉ, P f ) (Ĉ, P g) such that h is homotopic to ĥ rel P f and or f and g are Lattès maps. ĥ f = g ĥ, We now explain what Lattès maps are. For τ H = {z C Im z > 0}, let T τ be the complex torus C/(Z + τz). For n 2, the map z nz descends to a holomorphic map from T τ to itself. In fact, it further descends to a map L n,τ from the quotient T τ / z z Ĉ to itself, called a Lattès map. Note that the isomorphism T τ / z z Ĉ is given by the Weierstrass function. One can show that the post-critical set for L n,τ has 4 points. If τ 1 τ 2, then L n,τ1 and L n,τ2 are Thurston equivalent but not conformally conjugate. Sketch of proof for Theorem 3.1. Let h 0 : (Ĉ, P f ) (Ĉ, P g) be the Teichmüller map and lift it to a map h 1 : (Ĉ, P f ) (Ĉ, P g) such that h 0 f = g h 1. Then h 1 = h 0 by Teichmüller s uniqueness theorem.

4 4 LECTURE 2 If h 0 is not conformal, then there are associated initial and terminal foliations F i and F t on Ĉ with pronged singularities. We can form the orientation double covers S i and S t associated to these foliations. The lifted foliations are orientable and hence don t have 1-pronged singularities. There is an Euler characteristic formula for singular foliations which implies that S i and S t cannot be spheres and are thus complex tori. One can then deduce that f and g are Lattès maps. 4. Topological characterization of rational maps We now turn to the question of existence. Suppose that q : S 2 S 2 is a topological branched cover and P q <. When does there exist a holomorphic map f : Ĉ Ĉ which is postcritically finite and Thurston equivalent to q? The answer requires the notion of weighted multicurves. A multicurve for q is a collection of non-trivial, non-peripheral, non-homotopic, disjoint, simple, closed curves in S 2 \ P q. A trivial curve is one which bounds a disk and a peripheral curve is one which bounds a punctured disk. A weighted multicurve is a multicurve whose curves are given real positive weights. homotopic curves peripheral curve trivial curve Figure 5. Not a multicurve Figure 6. A multicurve

5 LECTURE 2 5 Given a weighted multicurve X = i I w iγ i, we define a new weighted multicurve q X as follows. The curves α j for q Γ are the connected components of the full preimage q 1 ( i I γ i) which are non-trivial and non-peripheral in S 2 \ P q. If q maps α j to γ i with degree d j, then α j is given weight w i /d j. If there is more than one curve in a given free homotopy class, we add their weights, assign this new weight to one of the curves, and discard the others. We write q X X if for every curve γ X, there is a curve α q X homotopic to γ in S 2 \ P q and the weight of α in q X is at least as large as the weight of γ in X. The answer to the above existence question is as follows : Theorem 4.1. Suppose that q : S 2 S 2 is a postcritically finite topological branched cover. Then either q is Thurston equivalent to a holomorphic map f : Ĉ Ĉ, or there exists a weighted multicurve X on S 2 \ P q such that q X X. 5. Conformal dynamical systems with finite topology Let U and V be disks with holes, let f : U V be a covering map, and let i : U V be an orientation-preserving embedding. U i f Given a complex structure σ on V, there is a unique complex structure on U such that f is holomorphic. We denote that complex structure by f σ. We define the Teichmüller space Teich(i, f) as the subset of Teich(V ) of (marked) complex structures σ on V such that there is a conformal embedding j : (U, f σ) (V, σ) homotopic to i. Such a conformal structure generates a (partial) conformal dynamical system, where starting with any point z 0 j(u), we study the sequence defined by z n+1 := f(j 1 (z n )), which makes sense as long as z n j(u). A first question we might ask is : When is Teich(i, f) non-empty? The answer is essentially contained in Thurston s theorem 4.1. Another question which we wish to address during this course is : When is Teich(i, f) compact? The answer will read as follows : Either Teich(i, f) is compact V

6 6 LECTURE 2 or there exists a weighted multiarc X on V which is dominated by i f X, which we denote by i f X X.