On the dynamics of composite entire functions

Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains. The second author was supported by Max-Planck-Gessellschaft ZFDW, and by Tian Yuan Foundation, NSFC.     

On the dynamics of entire functions on the Julia set

Let f be an entire function and ?ⁿ its n-th iterate. Let P(?) denote the postcritical set of ? and J(?) the Julia set of ?. Suppose that the set E of all z ∈ J(?) with limsup_n→∞ dist (?ⁿ(z), P(?) U {∞}) > 0 has positive measure. It is proved that for a given set A ? ? of positive measure the set {n ∈ ?; ?ⁿ(z) ∈ A} is infinite for almost all z in the plane. From this follows that the forward orbit of almost all z ∈ ? is dense in the plane if E has positive measure.     

On the proof of the reciprocity law for arithmetic Siegel modular functions

Earlier we obtained a new proof of Shimura’s reciprocity law for the special values of arithmetic Hilbert modular functions. In this note we show how from this result one may derive Shimura’s reciprocity law for special values of arithmetic Siegel modular functions. To achieve this we use Shimura’s classification of the special points of the Siegel space, Satake’s classification of the equivariant holomorphic imbeddings of Hilbert-Siegel modular spaces into a larger Siegel space, and, finally, a corrected version of some of Karel’s results giving an action of the Galois group Gal(Q_ab/Q) on arithmetic Siegel modular forms. Research supported in part by the NSF Grant No. DMS-8601130.     

拟圆周的两个几何性质

§1　IntroductionLetΓbe a Jordan curve of R2 and f∶R2→R2 be a k-quasiconformal mapping,where1≤k<+∞.Γis called a quasicirlce ifΓis the image of the unit circle B2 under f.It is well-known that quasicircles play a very important role in quasiconformalmapping theory,complex dynamics,Fuchsian groups,Teichmuller space theory and lowdimensional topology,( see[1—5] etc.)In1 963 ,Ahlfors obtained the three-point property of quasidisks[6] .Later,Gehring[7] ,Osgood[8] ,Krzyz[9] ,Ch…     

On quasiplanes in Euclidean spaces

A variational inequality for the images of -dimensional hyperplanes under quasiconformal maps of the -dimensional Euclidean space is proved when

     

On a class of prime entire functions

In this note it is proved that entire functions of the form Fa（z）=H（z）-aα（z）
where H andα are entire functions, with α having at least one zero, H＇ and α＇having no common zeros and T（r, α） = S（r, H）, are prime in entire sense for most values of a. The class of these functions is seen to contain examples of both periodic and non-periodic prime entire functions found by Noda, Qiao, Urabe and by Singh and Charak. Another result due to Singh and Charak will be improved as well.     

On a non-standard class of entire functions

In this article we construct a class of entire functions of arbitrary finite order and with non-zero Taylor coefficients, which is not written in the form of Hadamard's canonical product. Some basic properties of this class are also given.     

On a unit group generated by special values of Siegel modular functions

There has been important progress in constructing units and -units associated to curves of genus 2 or 3. These approaches are based mainly on the consideration of properties of Jacobian varieties associated to hyperelliptic curves of genus 2 or 3. In this paper, we construct a unit group of the ray class field of modulo 6 with full rank by special values of Siegel modular functions and circular units. We note that . Our construction of units is number theoretic, and closely based on Shimura's work describing explicitly the Galois actions on the special values of theta functions.

     

On the growth of entire functions

Suppose thatα>1, 0<R<∞ and thatf is analytic in |z|≤αR with |f(0)|≥1. It is shown that for a constant dα depending only onα, . Therefore iff is entire of order λ<∞, logM(r,f)/T(r,f) has order at most λ/2. These results are shown by example to be quite precise.     

Hausdorff dimension of Julia set of a polynomial with a Siegel disk

Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2.     

Continuity of Julia set for a family of entire functions

Based on the work of McMullen about the continuity of Julia set for rational functions, in this paper, we discuss the continuity of Julia set and its Hausdorff dimension for a family of entire functions which satisfy some conditions.     

Rigidity of escaping dynamics for transcendental entire functions

We prove an analog of Böttcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $ \mathcal{B} $ . More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane. We also prove that the conjugacy is essentially unique. In particular, we show that a function $ f \in \mathcal{B} $ has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions $ f,g \in \mathcal{B} $ that belong to the same parameter space are conjugate on their sets of escaping points.     

Semigroups of transcendental entire functions and their dynamics

We investigate the dynamics of semigroups of transcendental entire functions using Fatou–Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some condition for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.     

A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.