Existence of absorbing domains

Transcription

1 Existence of absorbing domains K. Barański, N. Fagella, B. Karpińska and X. Jarque Warsaw U., U. de Barcelona, Warsaw U. of Technology Universitat de Barcelona Bȩdlewo, Poland April 23, 2012 (Diada de Sant Jordi, Books an Roses )

2 The notion of absorbing domains Definition: Let U be a hyperbolic domain in C and let F : U U be a non constant holomorphic map. A domain W U is absorbing for F on U if F (W ) W, and for every compact set K U there exists n = n(k) > 0, such that F n (K) W. Example: If U is an invariant attracting basin and z 0 is its corresponding attracting fixed point (for F ) then We can take W to be a (simply connected) image of a (punctured) disc D centered at z = 0, and F in W {z 0 } is conformally conjugated to z f (z 0 )z in D. Remark: As in the example, W is usually a simply connected set.

3 Cowen s Theorem Theorem [Cowen, 1981]: Let G : H H be a holomorphic map such that G n as n (no fixed points: z 0 = ). Then there exists a simply connected domain V H, a domain Ω {H, C}, a holomorphic map ϕ : H Ω, and a Möbius transformation T mapping Ω onto itself, such that: (a) V is absorbing for G on H, (b) ϕ(v ) is absorbing for T on Ω, (c) ϕ G = T ϕ on H (i.e., ϕ is a semiconjugacy on H), (d) ϕ is univalent on V (i.e., ϕ is a conjugacy on V H). Moreover (up to a Möbius transformation) Ω = C, T (ω) = ω + 1, Ω = H, T (ω) = aω for some a > 1, Ω = H, T (ω) = ω ± i.

4 Cowen s Theorem In other words, we have the following (blue part) commutative diagram. T ϕ(v ) Ω Ω ϕ ϕ 1 V G ϕ H H π π U F U with Ω {C, H}. Moreover, {ϕ, T, Ω} depends only on (the speed to infinity of the orbits of) G.

5 König s Theorem (existence of simply connected absorbing domains) Theorem [König, 1999]: Let U be a hyperbolic domain in C and let F : U U be a holomorphic map, such that F n as n and for every closed curve γ U there exists n > 0 such that F n (γ) is contractible in U. Then there exists a simply connected domain W U, a domain Ω and a transformation T (as in Cowen s Theorem), and a holomorphic map ψ : U Ω, such that: (a) W is absorbing for F on U, (b) ψ(w ) is absorbing for T on Ω, (c) ψ F = T ψ on U (i.e., ψ is a semiconjugacy on U), (d) ψ is univalent on W (i.e., ψ is a conjugacy on W ).

6 König s Theorem In other words, we have the following commutative diagram. ϕ(v ) ϕ 1 V π T Ω Ω ϕ G ϕ H H π π W := π(v ) U F U where ϕ and V follows from Cowen s Theorem applied to G, a lift of F by a universal covering π : H U. In fact ψ = ϕ π 1 is well defined (Marden Pommerenke, 1980). Moreover π is univalent in V and W := π(v ) is simply connected.

7 König s Theorem and Baker domains Definition (of a Baker domain): Let f : C Ĉ transcendental meromorphic. A p periodic Fatou component U C is a Baker domain if for all z U we have f np (z) ζ U, and f k (ζ) is not defined for some k = 0,... p 1. Corollary [of König s Theorem]: Let f be a meromorphic map with finitely many poles. If U is a Baker domain, then there exists a simply connected absorbing domain W for F = f p on U. (In König s lenguage there is a conformal conjugacy for F = f p on U) Theorem [König, 1999]: There is a meromorphic function f with infinitely many poles whose Fatou set contains an invariant Baker domain U which does not admit a simply connected absorbing domain for F = f on U.

8 Existence of absorbing domains Theorem [Baranski,Fagella, Karpisnka and J.] Let U be a hyperbolic domain in C and let F : U U be a holomorphic map such that F n (z) as n for z U. Then there exists an absorbing domain W for F on U (in general W is not simply connected). Moreover (Notation: closures in C and D U (a, r) hyperbolic discs) (a) W U (b) F j (W ) = F j (W ) W, for j 1 (c) F n (W ) =, and n=1 (d) for every point z U and every sequence of positive numbers r n, n 0 with lim n r n =, the domain W can be chosen such that W D U (F n (z), r n ). n=0

9 Defining the set W through A (case Ω = H, T (z) = z + i) F (W ) W U F 2 (W )

10 Existence of absorbing domains and Baker domains Corollary: Let f : C C be a meromorphic map and let U be a periodic Baker domain for f. Then there exists an absorbing domain W satisfying the properties listed in the previous Theorem for the map F = f p. Remark: This corollary closes the problem of finding absorbing domains for the (Denjoy-Wolf) point z 0 associated to a periodic Fatou domain. That is, open neighborhoods (of z 0 ) for attracting periodic points, petals for parabolic periodic points, and W for Baker domains (with z 0 = and F not defined there). Of course the cases of wandering domains and Herman rings are not related with any Denjoy-Wolf s point.

11 The key idea (of the proof) ϕ(v ) ϕ 1 V π W := π(v ) U T Ω Ω ϕ ϕ G H H π π F U A ϕ 1 ϕ(v ) Ω Ω ϕ 1 ϕ ϕ ϕ 1 (A) V H π π π W := π ( ϕ 1 (A) ) π(v ) U T g H π F U König absorbing domain Our approach Remark: We should redefine König s absorbing domain W := π(v ) since otherwise we cannot guarantee, in general, that W U. Choosing A ϕ(v ), conveniently, is the right tool to (re)define W := π ( ϕ 1 (A) ) with the desired properties.

12 Defining the set A (case Ω = H, T (z) = z + i) A := n n 0 D H (T n (w), c n) T (ϕ(v )) T n 0 +3 (w) ϕ(v ) ( ) T D H (T n 0 +1 (w), c n0 +1) H T (z) = z + i H T n 0 (w). ( ) T D H (T n 0 (w), c n0 ) D H (T n 0 (w), c n0 ) ϕ(v ) w

13 Defining the set A (case Ω = H, T (z) = z + i) For any w H and any given sequence {b n } n 0, we want to define A := D H (T n (w), c n ) D ϕ(v ) (T n (w), b n ) ϕ(v ) n n 0 n n 0 with {c n } being increasing and tending to such that W = π ( ϕ 1 (A) ) A ϕ(v ) (W well defined) W absorbing (for F on U) A absorbing (for T on Ω) W U A ϕ(v ) not too big... F j (W ) = F j (W ) W T j ( A ) A, j 1 F n (W ) =... n=1

14 Defining the set W through A (case Ω = H, T (z) = z + i) πϕ 1 (A) r A := n n D 0 H (T n (w), c n) πϕ 1 T n 0 +3 (w) H. D H (T n 0 (w), c n0 ) ϕ(v ) πϕ 1 (r) U w A = n n 0 D H (T n (w), c n) U n n 0 D U (T n (z 0 ), b n)

15 Defining the set W through A (case Ω = H, T (z) = z + i) For any w H and any given sequence {b n } n 0, we want to define A := D H (T n (w), c n ) D ϕ(v ) (T n (w), b n ) ϕ(v ) n n 0 n n 0 with {c n } being increasing and tending to such that W = π ( ϕ 1 (A) ) A ϕ(v ) (W well defined) W absorbing (for F on U) A absorbing (for T on Ω) W U A ϕ(v ) not too big... F j (W ) = F j (W ) W T j ( A ) A, j 1 F n (W ) =... n=1

16 A is absorbing We should show that T (A) A. T (D H (T n (w), c n )) = D H (T n+1 (w), c n ) D H (T n+1 (w), c n+1 ), where the later inclusion follows from {c n } being increasing. For every compact set K H there exists n = n(k) > 0, such that T n (K) A Take K H compact. Clearly K D H (w, r) for some r > 0. So, T n (K) T n (D H (ω, r)) = D H (T n (ω), r) D H (T n (ω), c n ) A for sufficiently large n, because c n.

17 W is absorbing We should show that F (W ) W. (Indeed this follow from the fact that F j (W ) W ) For every compact set K U there exists n = n(k) > 0, such that F n (K) W (a) Take K U compact. Let u 1,..., u k K, such that k K π(n(w j )), j=1 π(w j ) = u j and N( ) denotes a neighborhood and N( ) H. (b) The set L = k j=1 ϕ(n(w j)) is compact in Ω. So, T n (L) A.

18 W is absorbing (c) This implies k k k G n (N(w j )) ϕ 1 T n (ϕ(n(w j ))) = ϕ 1 ϕ(g n (N(w j ))) ϕ 1 (A), j=1 j=1 j=1 k k F n (K) F n (π(n(w j ))) = π(g n (N(w j ))) πϕ 1 (A) := W, j=1 j=1 T n L A T n (L) ϕ G n ϕ ϕ 1 kj=1 (N(w j )) π K U

19 The end Dziȩkujȩ!! z okazji urodzin, Feliks!!