Interplay between polynomial and transcendental entire dynamics

Transcription

1 and Interplay and transcendental Department of Mathematical Sciences, University of Liverpool Oberwolfach, October 2008

2 Outline and 1 and 2 3 4

6 and I shall be talking today about some aspects of the of s functions that are related to each other. Influences of on transcendental : Directly applicable proofs/concepts. from successful ideas in. Phenomena special to transcendental. Influences of transcendental on.

12 Transcendental and f : C C, transcendental; f n := f f f } {{ } n times Fatou set F(f ): regular set. (Set of equicontinuity of the family (f n ).) Julia set: chaotic set. J(f ) := C \ F(f ). Escaping set J(f ) = I(f ). I(f ) := {z C : lim n f n (z) = }.

18 and z z 2 + c Some Julia sets Julia set J is compact. escaping set is the basin of infinity, contained in the Fatou set.

21 and z z 2 + c Some Julia sets z exp(z) 2

22 Some Julia sets and J is an uncountable union of Jordan arcs g : [0, ) C with g(t). We call g ( (0, ) ) a ray and g(0) the endpoint of that ray. set E of all endpoints has Hausdorff dimension 2, but the union R of all rays has Hausdorff dimension 1. E is totally disconnected, but E { } is connected. z exp(z) 2

26 Some Julia sets and set R of rays is completely contained in the escaping set I(f ). Some endpoints belong to I(f ); others do not. z exp(z) 2

27 Some Julia sets and set R of rays is completely contained in the escaping set I(f ). Some endpoints belong to I(f ); others do not. z exp(z) 2

28 Rays of exponential maps and Curves of escaping points for exponential maps will belong to the Julia set, but nonetheless provide a natural generalization of external rays of s. We will give a proof of the existence of these curves for the simple case f (z) = exp(z) 2 on the board. (This is not the original proof, due to Devaney!)

31 A question of Eremenko and Question Let f is a transcendental function and z I(f ). Can z be connected to infinity by a curve of escaping points of f? (re is also a related question about the existence of such curves, due to Fatou.) orem 1 (Schleicher,Zimmer) If f (z) = exp(z) + κ, κ C, then the answer is yes.

34 and

35 and Singular values set sing(f 1 ) contains all values in which some branch of f 1 cannot be defined. re are two types of such points: c is a critical value if c = f (w), f (w) = 0. a is an asymptotic value if there is a curve γ : (0, 1] C such that lim t 0 γ(t) = and lim t 0 f (γ(t)) = a. S(f ) := sing(f 1 ): singular values. Behaviour of singular orbits dominates.

37 and Singular values set sing(f 1 ) contains all values in which some branch of f 1 cannot be defined. re are two types of such points: c is a critical value if c = f (w), f (w) = 0. a is an asymptotic value if a f (γ) γ

40 B and A of functions which is particularly interesting for our considerations was introduced by Eremenko and Lyubich in 1986: B := {f transcendental, : sing(f 1 ) is bounded}. If f B, then I(f ) J(f ). In the following, we will mainly restrict ourselves to the.

43 An function f has finite order if Curves in I(f ) and log log f (z) = O(log z ) (z ). orem 2 (Rottenfußer, Rückert, R., Schleicher) Suppose that f B has finite order. n every escaping point can be connected to infinity by a curve of escaping points. ( theorem holds more generally for finite compositions of such functions.) Barański independently proved the same result for hyperbolic f with a single completely invariant Fatou component.

47 and orem 3 (RRRS) Counters re exists a hyperbolic function f B such that every path-connected component of J(f ) is bounded. function f can be chosen such that log log f (z) = (log z ) 1+ε. re are even hyperbolic functions f B such that every path-connected component of J(f ) is a point (but J(f ) is a compact connected set).

50 Adam Epstein s favorite and Let p be a of degree 2 with a repelling fixed point at 0, and µ := p (0). Let ψ be the (inverse of the) Kœnigs linearizing coordinate, defined near 0: ψ(µ z) = p(ψ(z)). Let Ψ : C C be the analytic extension of ψ to the complex plane (using the functional relation). (Ψ is called a Poincaré function.)

54 Adam Epstein s favorite and set of singular values of Ψ is determined by the postsingular set of p: S(Ψ) = n 1 p n (Crit(p)) So if the Julia set of p is connected, then Ψ B. We also note that Ψ has finite order of growth.

57 Adam Epstein s favorite and

58 An analog of Böttcher s theorem We say that f, g B are quasiconformally equivalent near if there are quasiconformal maps φ, ψ : C C such that and ψ(f (z)) = g(φ(z)) whenever f (z) or g(z) is sufficiently large. orem 4 (R., 2005) Suppose that f and g are quasiconformally equivalent near. n there is R > 0 and a quasiconformal map θ : C C such that θ f = g θ on J R (f ) := {z C : f n (z) R for all n 1}. Furthermore, the complex dilatation of θ on I(f ) J R (f ) is zero.

61 and

62 Consequences and result seems surprising given the existence of very wild s in B. It gives rise to the hope that escaping sets can also be used successfully to study the Julia sets of such wild functions. It also gives some explanation as to why the Julia sets of different explicit functions often look remarkably similar.

65 Interplay and Similarities in Julia sets z 7 2(exp(z) 1) z 7 λ sinh(z) z 7 (z+1) exp(z) 1

66 Rigidity results and conjugacy θ from the preceding theorem is essentially unique (up to countably many choices). orem 5 (R., 2005) Let f B. n there are no invariant line fields on I(f ). This and other rigidity results on the escaping set are used in work with van Strien on rigidity and density of hyperbolicity in families of real transcendental functions.

69 Transcendental s? and Aspects of transcendental have started to become apparent in renormalization phenomena (i.e. when the degree gets large). Features of exponential appearing in parabolic renormalization (Shishikura). Trancendental aspects of measurable (Urbanski-Zdunik) appearing in work of Avila-Lyubich on Feigenbaum quadratics.

70 Cantor bouquets and hedgehogs and proofs of the above results on existence of curves and conjugacy can be adapted to deal with model hedgehogs and (in the first case) actual hedgehogs. (This is ongoing work by Shishikura / Buff-Chéritat-Inou-R.)

71 A Siegel hedgehog and

72 A Siegel hedgehog and