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Introduction to Holomorphic Dynamics: Very Brief Notes

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Introduction to Holomorphic Dynamics: Very Brief Notes Introduction to Holomorphic Dynamics: Very Brief Notes Saeed Zakeri Last modified: 5-8-2003 Lecture 1. (1) A Riemann surface X is a connected complex manifold of dimension 1. This means that X is a connected Hausdorff space, locally homeomorphic to R2, which is equipped with a complex structure f(Ui; zi)g. Here each Ui is an open subset of X, S =» the union Ui is X, each local coordinate zi : Ui ¡! D is a homeomorphism, and ¡1 whenever Ui \ Uj 6= ; the change of coordinate zjzi : zi(Ui \ Uj) ! zj(Ui \ Uj) is holomorphic. In the above definition we have not assumed that X has a countable basis for its topology. But this is in fact true and follows from the existence of a complex structure (Rado’s Theorem). (2) A map f : X ! Y between Riemann surfaces is holomorphic if w ± f ± z¡1 is a holomorphic map for each pair of local coordinates z on X and w on Y for which this composition makes sense. We often denote this composition by w = f(z).A holomorphic map f is called a biholomorphism or conformal isomorphism if it is a homeomorphism, in which case f ¡1 is automatically holomorphic. (3) Examples of Riemann surfaces: The complex plane C, the unit disk D, the Riemann b sphere C, complex tori T¿ = C=(Z © ¿Z) with Im(¿) > 0, open connected subsets of Riemann surfaces such as the complement of a Cantor set in C. (4) The Uniformization Theorem: Every simply connected Riemann surface is confor- mally isomorphic to Cb, C, or D. (5) Classical form of Schwarz Lemma: If f : D ! D is holomorphic and f(0) = 0, then jf 0(0)j · 1. If jf 0(0)j = 1, then f is a rigid rotation around the origin. 0 " (6) If f : D(p; ±) ! D(q; ") is holomorphic, then jf (p)j · ± . (7) Corollary (Liouville): Every bounded holomorphic function C ! C must be con- stant. Lecture 2. (1) The automorphism group of a Riemann surface X is the group of all conformal isomorphisms X ! X. It is denoted by Aut(X). 1 2 (2) Theorem: az + b Aut(Cb) = fz 7! : a; b; c; d 2 C and ad ¡ bc 6= 0g »= PSL (C) cz + d 2 In particular, Aut(Cb) is a connected complex Lie group of dimension 3, homeo- morphic to the product RP3 £ R3 (this follows, for example, from the Iwasawa Decomposition). (3) Theorem: Aut(C) = fz 7! az + b : a; b 2 C and a 6= 0g Thus, Aut(C) can be identified with the subgroup of Aut(Cb) consisting of the maps which fix the point at infinity. It follows that Aut(C) is a connected complex Lie group of dimension 2, homeomorphic to the product C¤ £ C. (4) Theorem: µ ¶ z ¡ a Aut(D) = fz 7! ¸ : a 2 D and ¸ 2 C with j¸j = 1g 1 ¡ az Thus, Aut(D) can be identified with the identity component of the subgroup of b 1 Aut(C) consisting of the maps which commute with the reflection z 7! z . In particular, Aut(D) is a connected real Lie group of dimension 3, homeomorphic to the product D £ S1. (5) Theorem: az + b Aut(H) = fz 7! : a; b; c; d 2 R and ad ¡ bc > 0g »= PSL (R) cz + d 2 Thus, Aut(H) can be identified with the identity component of the subgroup of Aut(Cb) consisting of the maps which commute with the reflection z 7! z. (6) The action of Aut(Cb) on Cb is simply 3-transitive. Similarly, the action of Aut(C) on C is simply 2-transitive. The action of Aut(D) on D is transitive but not simply transitive. (7) Every non-identity σ 2 Aut(Cb) has two fixed points counting multiplicities. If σ has a double fixed point, it can be conjugated to the translation z 7! z + 1. In this case we call it parabolic. If σ has two distinct fixed points, it can be conjugated to the linear map z 7! ¸z for some ¸ 2 C r f0; 1g. The pair f¸; ¸¡1g is uniquely determined by σ. We call σ elliptic if j¸j = 1, hyperbolic if ¸ 2 R and j¸j 6= 1, and loxodromic otherwise. b 2 (8) An element σ 2 Aut(C) can be thought of as a matrix in PSL2(C), so ¿ = tr (σ) is well-defined. Then, σ is parabolic if ¿ = 4, elliptic if ¿ 2 [0; 4[, hyperbolic if ¿ 2 ]4; +1[, and loxodromic if ¿ 2 C r [0; +1[. 3 (9) Let σ; º be non-identity elements in Aut(Cb). If Fix(σ) = Fix(º), then σº = ºσ. Conversely, if σº = ºσ, then Fix(σ) = Fix(º) unless σ and º are involutions, each interchanging the two fixed points of the other (such as the commuting pair 1 σ(z) = ¡z and º(z) = z ). (10) Corollary: Two non-identity elements of Aut(C) or Aut(D) commute if and only if they have the same fixed point set. Lecture 3. (1) Review of covering space theory: ² Let X be a connected finite dimensional manifold. There exists a covering space ¼ : Xe ! X, with Xe simply-connected, called the universal covering of X. It is unique up to isomorphism of coverings. e ² The deck group of ¼ : X ! X, denoted by ΓX , consists of all homeomor- e e phisms γ : X ! X which satisfy ¼γ = ¼. Algebraically, ΓX is isomorphic to the fundamental group ¼1(X). Once a base point x 2 X is chosen, an iso- morphism between ΓX and ¼1(X; x) can be defined by sending γ 2 ΓX to the homotopy class of the projection of any path joining some x˜ 2 ¼¡1(x) to γ(˜x). e ² ΓX acts simply transitively on the fibers of ¼: If x;˜ y˜ 2 X with ¼(˜x) = ¼(˜y), there exists a unique γ 2 ΓX such that γ(˜x) =y ˜. In particular, if γ 2 ΓX has a fixed point, then γ = id. e e ² ΓX acts evenly on X: Every point in X has a neighborhood U such that γ(U)\ U = ; for all γ 2 ΓX r fidg. In particular, ΓX is a discrete subgroup of the e e homeomorphism group of X. The quotient X=ΓX is a Hausdorff manifold homeomorphic to X. There is a one-to-one correspondence between subgroups of ΓX and coverings of X as follows. e ² Given a subgroup H ½ ΓX , the quotient Y = X=H is a covering of X, with the covering map p : Y ! X defined by sending the H-orbit of x˜ 2 Xe to the » e ΓX -orbit of x˜. For this covering, ¼1(Y ) = H and the projection X ! Y is the universal covering. The deck group of p : Y ! X is isomorphic to N(H)=H, where ¡1 N(H) = fγ 2 ΓX : γHγ ½ Hg is the normalizer of H in ΓX . 4 ² Conversely, given any covering p : Y ! X, there exists a covering map e q : X ! Y such that pq = ¼. Moreover, there exists a subgroup H ½ ΓX e isomorphic to ¼1(Y ) such that X=H is homeomorphic to Y . (2) If X is a Riemann surface, the topological universal covering ¼ : Xe ! X can be equipped with the pull-back complex structure so as to make Xe into a Riemann e surface, ¼ into a holomorphic map, and ΓX into a subgroup of Aut(X). (3) Corollary: Every Riemann surface X can be represented as X=e Γ, where Xe is con- formally isomorphic to Cb, C or D, and Γ is a subgroup of Aut(Xe) isomorphic to e ¼1(X) which acts evenly on X. (4) A Riemann surface X is called spherical, Euclidean, or hyperbolic according as its universal covering Xe is conformally isomorphic to Cb, C or D. ² Xe »= Cb. Since every automorphism of Cb has a fixed point, the only subgroup of Aut(Cb) which acts evenly on Cb is the trivial group. It follows that X »= Cb. ² Xe »= C. The only fixed point free automorphisms of C are translations. It easily follows that the only subgroups of Aut(C) which act evenly are the trivial group, or the group generated by a single translation z 7! z + b, or the b1 group generated by two translations z 7! z + b1 and z 7! z + b2, with 2= R. b2 It follows that X »= C, or X »= C¤, or X »= a complex torus. ² Xe »= D. All other Riemann surfaces are therefore in this category. In partic- ular, a Riemann surface with non-abelian fundamental group must be hyper- bolic. (5) Examples: The punctured disk D¤ and the annuli A(1;R) = fz : 1 < jzj < Rg are hyperbolic. In fact, these are the only hyperbolic Riemann surfaces with non-trivial abelian fundamental group. The trice punctured sphere Cb r fa; b; cg is hyperbolic since its fundamental group is non-abelian. If we assume fa; b; cg = f0; 1; 1g (a normalization which can always be achieved by applying an automorphism of Cb), an explicit universal covering map is given by the elliptic modular function H ! C r f0; 1g. (6) Let f : X ! Y be a holomorphic map between Riemann surfaces. If X is non- hyperbolic and Y is hyperbolic, then f is constant. (7) Corollary (Picard): An entire function which omits two distinct values is constant. (8) Corollary: A domain X ½ Cb is hyperbolic iff Cb r X has at least three points. Lecture 4. 5 (1) Let p 2 D, v 2 TpD and choose ' 2 Aut(D) so that '(p) = 0.
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