A wandering domain in class B on which all iterates are univalent

Transcription

1 A wandering domain in class B on which all iterates are univalent Núria Fagella (with X. Jarque and K. Lazebnik) Facultat de Matemàtiques i Informàtica Universitat de Barcelona and Barcelona Graduate School of Mathematics Workshop on Complex Dynamics 2017 December 11-15, 2017 N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 1 / 32

2 Transcendental dynamics If f : C C (or to Ĉ) has an essential singularity at infinity we say that f is transcendental. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 2 / 32

3 Transcendental dynamics If f : C C (or to Ĉ) has an essential singularity at infinity we say that f is transcendental. Transcendental maps may have Fatou components that are not basins of attraction nor rotation domains: U is a Baker domain of period 1 if f n U loc. unif. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 2 / 32

4 Transcendental dynamics If f : C C (or to Ĉ) has an essential singularity at infinity we say that f is transcendental. Transcendental maps may have Fatou components that are not basins of attraction nor rotation domains: U is a Baker domain of period 1 if f n U loc. unif. U is a wandering domain if f n (U) f m (U) = for all n m. z + a + b sin(z) [Figures: Christian Henriksen] z + 2π + sin(z) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 2 / 32

5 Transcendental dynamics The set S(f ) of singularities of f 1 consists of critical values and asymtpotic values (and the closure of such). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 3 / 32

6 Transcendental dynamics The set S(f ) of singularities of f 1 consists of critical values and asymtpotic values (and the closure of such). A critical value is the image of a critical point, i.e. v = f (c) with f (c) = 0. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 3 / 32

7 Transcendental dynamics The set S(f ) of singularities of f 1 consists of critical values and asymtpotic values (and the closure of such). A critical value is the image of a critical point, i.e. v = f (c) with f (c) = 0. A point a Ĉ is an asymptotic value if there exists a curve γ(t) such that f (γ(t)) a. (Morally, a has infinitely many preimages collapsed at infinity). Example: z = 0 for f (z) = λe z. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 3 / 32

8 Transcendental dynamics The set S(f ) of singularities of f 1 consists of critical values and asymtpotic values (and the closure of such). A critical value is the image of a critical point, i.e. v = f (c) with f (c) = 0. A point a Ĉ is an asymptotic value if there exists a curve γ(t) such that f (γ(t)) a. (Morally, a has infinitely many preimages collapsed at infinity). Example: z = 0 for f (z) = λe z. f : C \ f 1 (S(f )) C \ S(f ) is a covering map of infinite degree. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 3 / 32

9 Transcendental dynamics The set S(f ) of singularities of f 1 consists of critical values and asymtpotic values (and the closure of such). A critical value is the image of a critical point, i.e. v = f (c) with f (c) = 0. A point a Ĉ is an asymptotic value if there exists a curve γ(t) such that f (γ(t)) a. (Morally, a has infinitely many preimages collapsed at infinity). Example: z = 0 for f (z) = λe z. f : C \ f 1 (S(f )) C \ S(f ) is a covering map of infinite degree. Define the postsingular set of f as P(f ) = s S n 0 f n (s). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 3 / 32

10 The Eremenko-Lyubich class Example: z λ Special classes B = {f ETF (or MTF) such that S(f ) is bounded} z sin(z). Maps in class B have no Baker domains and NO ESCAPING WANDERING DOMAINS (Escaping set J(f )). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 4 / 32

11 The Eremenko-Lyubich class Example: z λ Special classes B = {f ETF (or MTF) such that S(f ) is bounded} z sin(z). Maps in class B have no Baker domains and NO ESCAPING WANDERING DOMAINS (Escaping set J(f )). If U is a wandering domain, and L(U) is the set of limit functions of f n on U, then, all limit functions are constant and escaping if L(U) = { } U is oscillating if {, a} L(U) for some a C. bounded if / L(U). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 4 / 32

12 Existence of wandering domains Question Can maps in class B have wandering domains at all? N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 5 / 32

13 Existence of wandering domains Question Can maps in class B have wandering domains at all? Answer: yes. Theorem (Bishop 15) There exists an entire map f B such that f has an (oscillating) wandering domain. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 5 / 32

14 Existence of wandering domains Question Can maps in class B have wandering domains at all? Answer: yes. Theorem (Bishop 15) There exists an entire map f B such that f has an (oscillating) wandering domain. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 5 / 32

15 Univalent WD in class B Bishop s wandering domain contains critical points inside of arbitrarily high order. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 6 / 32

16 Univalent WD in class B Bishop s wandering domain contains critical points inside of arbitrarily high order. Question Is there an oscillating wandering domain with no singular values inside any of its iterates? N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 6 / 32

17 Univalent WD in class B Bishop s wandering domain contains critical points inside of arbitrarily high order. Question Is there an oscillating wandering domain with no singular values inside any of its iterates? Answer: yes. Theorem A (F-Lazebnik-Jarque 17) There exists an ETF f B such that f has a wandering domain U on which f n U is univalent for all n 0. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 6 / 32

18 Constructing entire functions Bishop s qc-folding construction Let f : C C be a transcendental entire function with exactly two critical values, say 1 and +1 no finite asymptotic values Question: What does f look like? N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 7 / 32

19 Bishop s qc-folding construction T = f 1 ([ 1, +1]) is an infinite bipartite tree. C T f C N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 8 / 32

20 Bishop s qc-folding construction T = f 1 ([ 1, +1]) is an infinite bipartite tree. C T f C H r i(2k+1)π i(2k)π cosh cosh : H r C\[ 1, +1] is a universal cover. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 8 / 32

21 Bishop s qc-folding construction T = f 1 ([ 1, +1]) is an infinite bipartite tree. C T f C Ω H r τ i(2k+1)π i(2k)π cosh Ω c.c. of C\T, τ Ω = (cosh 1 f Ω ) : Ω H r is conformal. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 8 / 32

22 Bishop s qc-folding construction Conversely: How to construct f from (T, τ)? More precisely: Given an infinite bipartite tree T C with good enough geometry a map τ such that τ Ω : Ω H r is conformal, Ω c.c. of C\T N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 9 / 32

23 Bishop s qc-folding construction Conversely: How to construct f from (T, τ)? More precisely: Given an infinite bipartite tree T C with good enough geometry a map τ such that τ Ω : Ω H r is conformal, Ω c.c. of C\T Question: Does there exist an entire function f : C C such that f = cosh τ? Main problem: In general, cosh τ is not continuous across T. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 9 / 32

24 Bishop s qc-folding construction Strategy: Step 1: Modify (T, τ) in a small neighborhood T (r 0 ) of T. More precisely, replace (T, τ) by (T, η) such that T T T (r 0 ) η = τ off T (r 0 ) η Ω : Ω H r is K-quasiconformal, Ω c.c. of C\T cosh η continuous across T (quasiregular map) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 10 / 32

25 Strategy: Bishop s qc-folding construction Step 1: Modify (T, τ) in a small neighborhood T (r 0 ) of T. More precisely, replace (T, τ) by (T, η) such that T T T (r 0 ) η = τ off T (r 0 ) η Ω : Ω H r is K-quasiconformal, Ω c.c. of C\T cosh η continuous across T (quasiregular map) Step 2: Apply MRMT. Obtain a qc map φ (the integrating map of µ cosh η ) so that f := cosh η φ 1 is entire. In particular f φ = cosh τ off T (r 0 ) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 10 / 32

26 C Bishop s qc-folding construction T H r Ω τ N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 11 / 32

27 C Bishop s qc-folding construction η H r T(r ) 0 Ω τ N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 11 / 32

28 C Bishop s qc-folding construction η H r T(r ) 0 Ω τ C cosh N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 11 / 32

29 C Bishop s qc-folding construction η H r T(r ) 0 Ω τ C ϕ ϕ(t) C cosh ϕ(t') N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 11 / 32

30 C Bishop s qc-folding construction η H r T(r ) 0 Ω τ C ϕ ϕ(t') ϕ(t) f C cosh N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 11 / 32

31 Bounded geometry Definition: We say that T has bounded geometry if edges of T are C 2 with uniform bounds angles between adjacent edges are uniformly bounded away from 0 1 diam(e) e, f adjacent edges, M diam(f ) M e, f non-adjacent edges, diam(e) dist(e,f ) M e z dist(z,e) diam(e) =r T(r) T (r) = e edge of T { } z C / dist(z, e) < r diam(e) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 12 / 32

32 Bounded geometry Lemma If T has bounded geometry, then r 0 > 0 such that Ω c.c. of C\T, square Q H r that has ) a τ Ω -edge as one side, Q τ Ω (T (r 0 ) Ω C H r τ T(r ) 0 Ω Every edge has two τ-sizes!!! N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 13 / 32

33 Bishop s Theorem Theorem (Bishop 12) If (T, τ) satisfies the following conditions 1 T has bounded geometry 2 every edge has τ-size π then an entire function f and a quasiconformal map φ such that f φ = cosh τ off T (r 0 ) Moreover f has exactly two critical values, 1 and +1 f has no finite asymptotic values φ(t ) f 1 ([ 1, +1]) (= φ(t )) c critical point of f, deg loc (c, f ) = deg(c, φ(t )) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 14 / 32

34 Bishop s qc-folding construction The main technical difficulty is to find a quasiconformal map ψ from a square to itself such that { ψ maps the left side to an edge of length π ψ is the identity on the right side ψ 3π π N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 15 / 32

35 Bishop s qc-folding construction The main technical difficulty is to find a quasiconformal map ψ from a square to itself such that { ψ maps the left side to an edge of length π ψ is the identity on the right side Solution: Add some extra edges and unfold. ψ 3π π N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 15 / 32

36 ψ 1 is called a quasiconformal folding. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 15 / 32 Bishop s qc-folding construction The main technical difficulty is to find a quasiconformal map ψ from a square to itself such that { ψ maps the left side to an edge of length π ψ is the identity on the right side Solution: Add some extra edges and unfold. ψ 3π π

37 Bishop s qc-folding construction The main technical difficulty is to find a quasiconformal map ψ from a square to itself such that { ψ maps the left side to an edge of length π ψ is the identity on the right side Solution: Add some extra edges and unfold. 13π W ψ π N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 15 / 32

38 ψ 1 is called a quasiconformal folding. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 15 / 32 Bishop s qc-folding construction The main technical difficulty is to find a quasiconformal map ψ from a square to itself such that { ψ maps the left side to an edge of length π ψ is the identity on the right side Solution: Add some extra edges and unfold. 13π W ψ π

39 Bishop s qc-folding construction adding singular values Generalization: We may also construct f with more critical values than only 1 and +1 some finite asymptotic values arbitrary high degree critical points Let T be an infinite bipartite graph. C R L R C R D R f D R R L N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 16 / 32

40 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

41 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω τ Ω H r D L Ω Ω τ Ω τ Ω D H l N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

42 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R D L Ω Ω Ω τ Ω cosh H r τ Ω τ Ω D H l N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

43 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω D (Ω, ) L Ω τ Ω cosh H r C\[ 1, +1] τ Ω (D, 0) τ Ω H l N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

44 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω D (Ω, ) L Ω τ Ω cosh H r C\[ 1, +1] τ Ω (D, 0) τ Ω H l z z d Ω (D, 0) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

45 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω D (Ω, ) L (Ω, ) τ Ω cosh H r C\[ 1, +1] τ Ω (D, 0) τ Ω (H l, ) z z d Ω (D, 0) ρ Ω (D, wω ) where ρ Ω : D D is quasiconformal with ρ Ω (z) = z, z D. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

46 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω D (Ω, ) L (Ω, ) τ Ω cosh H r C\[ 1, +1] τ Ω (D, 0) τ Ω (H l, ) z z d Ω (D, 0) exp (D, 0) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

47 Bishop s qc folding construction - adding singular values The c.c. of C\T are sorted into three different types: R-components: τ Ω : Ω H r conformally D-components: τ Ω : Ω D conformally L-components: τ Ω : Ω H l conformally R Ω D (Ω, ) L (Ω, ) τ Ω cosh H r C\[ 1, +1] τ Ω (D, 0) τ Ω (H l, ) z z d Ω (D, 0) exp (D, 0) ρ Ω (D, vω ) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 17 / 32

48 Constructiong the oscillating wandering domains in class B f = F φ 1 with { F : C C quasiregular (transcendental) φ : C C quasiconformal so that φ µ 0 = F (µ 0 ) F is constructed using an infinite graph. D 1 D 2 D 3 D 4 D 5 S + N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 18 / 32

49 Oscillating wandering domains in class B F : S + λ sinh cosh H r C\[ 1, +1] z cosh(λ sinh(z)) λsinh S H r cosh C λ > 0 is fixed so that f n ( 1 2) + very fast. n + N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 19 / 32

50 Oscillating wandering domains in class B F : S + λ sinh cosh H r C\[ 1, +1] z cosh(λ sinh(z)) λsinh S H r cosh C 0 0 λ > 0 is fixed so that f n ( 1 2) + very fast. n + N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 19 / 32

51 For every n 1, Oscillating wandering domains in class B with z (z z n) dn ρ n F : (D n, z n ) (D, 0) ( (D, wn ) z ρ n (z zn ) dn) ρ n : D D quasiconformal ρ n (0) = w n (z- z ) d n D n n ρ n z n 0 w n 1/2 for some parameters d n n + + and w n n N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 20 /

52 For every n 1, Oscillating wandering domains in class B with z (z z n) dn ρ n F : (D n, z n ) (D, 0) ( (D, wn ) z ρ n (z zn ) dn) ρ n : D D quasiconformal ρ n (0) = w n supp(µ ρn ) { 1 2 z 1} (z- z ) d n D n n ρ n z n 0 w n 1/2 for some parameters d n n + + and w n n N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 20 /

53 Oscillating wandering domains in class B Using Bishop s construction F may be extended to a quasiregular map F : C C such that: z C, F ( z) = F (z) and F (z) = F (z) Crit(F ) = { 1, +1} {w n, n 1} { 1 Asym(F ) = 2} D with wn n F D 1 D 2 D 3 D 4 D 5 S N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 21 / 32

54 Oscillating wandering domains in class B Using Bishop s construction F may be extended to a quasiregular map F : C C such that: z C, F ( z) = F (z) and F (z) = F (z) Crit(F ) = { 1, +1} {w n, n 1} { 1 Asym(F ) = supp (F (µ 0 )) is small enough 2} D with wn n F D 1 D 2 D 3 D 4 D 5 S N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 21 / 32

55 Oscillating wandering domains in class B Using Bishop s construction F may be extended to a quasiregular map F : C C such that: z C, F ( z) = F (z) and F (z) = F (z) Crit(F ) = { 1, +1} {w n, n 1} { 1 2} D with wn Asym(F ) = supp (F (µ 0 )) is small enough in order that φ S + Id S + n F D 1 D 2 D 3 D 4 D 5 S Let f = F φ with φ : C C quasiconformal so that µ φ 1 = F (µ 0 ). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 21 / 32

56 Oscillating wandering domains in class B ~ ~ D n D n+1 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

57 Oscillating wandering domains in class B f ~ ~ D n D n+1 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

58 Oscillating wandering domains in class B f ~ 1_ ~ ~ D n 4 D n D n+1 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

59 Oscillating wandering domains in class B f ~ 1_ ~ ~ D n 4 D n D n+1 U n 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) f n (U n ) = 1 4 D n and inradius(u n ) C. ( df n dx x= 1 2 ) 1 N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

60 Oscillating wandering domains in class B ~ ~ D n D n+1 ~ 1_ D 4 n+1 U n+1 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) f n (U n ) = 1 4 D n and inradius(u n ) C. f n+1 (U n+1 ) = 1 4 D n+1 and inradius(u n+1 ) C. ( df n ) 1 dx x= 1 ( 2 df n+1 ) 1 dx x= 1 2 N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

61 Oscillating wandering domains in class B f ~ 1_ ~ ~ D n 4 D n D n+1 U n+1 1/2 f(1/2) f n f n-1 (1/2) (1/2) S + n+1 f (1/2) ( df n+1 dx ) dn f n+1 (U n+1 ) = 1 D 4 n+1 and inradius(u n+1 ) C. ( w n f 1 D ) ( ( 4 n and diam f 1 D )) 4 n C. ( 1 4 x= 1 2 ) 1 N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 22 / 32

62 Oscillating wandering domains in class B This is an iterative process, where Bishop s theorem is applied infinitely many times. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 23 / 32

63 Oscillating wandering domains in class B This is an iterative process, where Bishop s theorem is applied infinitely many times. The parameters w n, d n are adjusted successively obtaining a new map f, everytime closer to the previous one, converging to a final map f. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 23 / 32

64 Oscillating wandering domains in class B This is an iterative process, where Bishop s theorem is applied infinitely many times. The parameters w n, d n are adjusted successively obtaining a new map f, everytime closer to the previous one, converging to a final map f. The correction map φ at every step is closer and closer to the identity, since the support of µ decreases exponentially. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 23 / 32

65 Modifying F to construct a univalent wandering domain The new map on D components Consider the map D D z z m + δz m 1 critical points which tend to D when m m 1 critical values which tend to δ in modulus when m. Let γ = { z = 3 2 } and then int(γ 1 ) contains all critical points. γ 1 γ N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 24 / 32

66 The new map on D components Define a quasiregular map ψ on D {z m + δz, if z int ( γ 1) ψ m = z m if z D, and linear interpolation in between int ( γ 1) and D. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 25 / 32

67 The new map on D components Define a quasiregular map ψ on D {z m + δz, if z int ( γ 1) ψ m = z m if z D, and linear interpolation in between int ( γ 1) and D. Claim 1 The dilatation of ψ m is uniformly bounded independently of m and δ << 1 2. supported on a region whose area tends to 0 as m. Idea of the proof N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 25 / 32

68 General strategy We resemble Bishop s construction with the following modifications: N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

69 General strategy We resemble Bishop s construction with the following modifications: Parameters λ, (w n ), (δ n ), (d n ). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

70 General strategy We resemble Bishop s construction with the following modifications: Parameters λ, (w n ), (δ n ), (d n ). Keep g(z) = cosh(λ sinh(z)) in the strip S +. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

71 General strategy We resemble Bishop s construction with the following modifications: Parameters λ, (w n ), (δ n ), (d n ). Keep g(z) = cosh(λ sinh(z)) in the strip S +. In the disks D n, substitute (z z n ) dn by ψ n (z z n ) := ψ dn (z z n ). N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

72 General strategy We resemble Bishop s construction with the following modifications: Parameters λ, (w n ), (δ n ), (d n ). Keep g(z) = cosh(λ sinh(z)) in the strip S +. In the disks D n, substitute (z z n ) dn by ψ n (z z n ) := ψ dn (z z n ). Compose with ρ n sending 0 to w n near 1 2. The critical values of ρ n φ n (z z n ) are now centered around w n. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

73 General strategy We resemble Bishop s construction with the following modifications: Parameters λ, (w n ), (δ n ), (d n ). Keep g(z) = cosh(λ sinh(z)) in the strip S +. In the disks D n, substitute (z z n ) dn by ψ n (z z n ) := ψ dn (z z n ). Compose with ρ n sending 0 to w n near 1 2. The critical values of ρ n φ n (z z n ) are now centered around w n. Bishop s theorem yields a quasiregular extension of g in C with dilatation independent of the parameters, and an entire map f = g φ, with φ qc. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 26 / 32

74 General strategy Now, as in Bishop, we modify g on the discs D n (adjust parameters d n, w n, δ n ), for n larger at each step. We iterate the process making sure that N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 27 / 32

75 General strategy Now, as in Bishop, we modify g on the discs D n (adjust parameters d n, w n, δ n ), for n larger at each step. We iterate the process making sure that the process converges N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 27 / 32

76 General strategy Now, as in Bishop, we modify g on the discs D n (adjust parameters d n, w n, δ n ), for n larger at each step. We iterate the process making sure that the process converges the modifications do not change the dynamics we want to preserve. N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 27 / 32

77 Exponents and dilatation parameters We consider straight annuli around D n (the safety belt) A n = {1 µ n < z z n < 1 + µ n }. Ã n D n, µn 2 µ n C n C n (cosh λ sinh φ 0 ) n Claim 2 radius(c n) radius(c n ) 1 + µ n n 1 µ n N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 28 / 32

78 First step We adjust µ 2 << µ 1, w 1 = f n 2 ( z n ) and δ 1 appropriately so that: D2, µ2 D 1, µ1 2 µ 1 Ã n2 2 µ 2 f n f 1 w 1 f 1 ( D 1, µ1 ) (f 1 ) n2 ( D n2 ) Crit. val. outside (f 1 ) n2 (Ãn 2 ) N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 29 / 32

79 Claim 3 The final check There exists a wandering domain U containing D 1, µ1, and it satisfies U n D. In particular, its orbit does not contain singular values and therefore f n U is univalent for all n. 3π 2π π 0 π 2π W 1 W 0 W 2 ˆτ V φ η 1 η 2 N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 30 / 32

80 Thank you for your attention! N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 31 / 32

81 Uniformly constant dilatation γ 1 w = m 1 Q 1 1 ψ m z m log log Estimate modq (right) and mod(ψ m (Q)) (left) and see that the quotient is uniformly bounded when m and δ 0. Go back N. Fagella, X. Jarque and K. Lazebnik (UB) Univalent WD RIMS Kyoto 32 / 32