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.     

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.     

On truncations of entire transcendental functions

In this paper it is shown that outside every circle, every finite value is assumed by some truncation of the Taylor series of an entire transcendental function (in fact, by every truncation of sufficiently high degree). This result differs from both Casorati-Weierstrass and Picard theorems inasmuch as the truncations actually assume the preassigned values and that there are no exceptional values.     

Growth of transcendental entire functions on algebraic varieties

LetX be a complete intersection algebraic variety of codimensionm>1 in ℂ ^m+n . We define the notion of (p,q)-order and (p,q)-K-type for transcendental entire functionsfεO(ℂ ^m+n ) whereK is a non-pluripolar compact subset of ℂ ^m+n . Further, we consider the analogues of (p,q)-order and (p,q)-K-type inO(X). We discuss the series expansions of the functions inO(X) in terms of an orthogonal basis in a Hilbert spaceL ²(X, μ), where μ is a capacitary extremal measure onK. Author is grateful to the NSA for partial support during the period of this research.     

Random dynamics of transcendental functions

This work concerns random dynamics of hyperbolic entire and meromorphic functions of finite order whose derivative satisfies some growth condition at ∞. This class contains most of the classical families of transcendental functions and goes much beyond. Based on uniform versions of Nevanlinna’s value distribution theory, we first build a thermodynamical formalism which, in particular, produces unique geometric and fiberwise invariant Gibbs states. Moreover, spectral gap property for the associated transfer operator along with exponential decay of correlations and a central limit theorem are shown. This part relies on our construction of new positive invariant cones that are adapted to the setting of unbounded phase spaces. This setting rules out the use of Hilbert’s metric along with the usual contraction principle. However, these cones allow us to apply a contraction argument stemming from Bowen’s initial approach.     

On the best polynomial approximation of entire transcendental functions in Banach spaces. II

We study the behavior of the best approximations of entire transcendental functionsf(z) of the order =0 by polynomials of at mostn th degree in the metric of the spaceE _p(),p1. In particular, we describe the relationship between the best approximations and the logarithmic order _L and type _L of the functionf(z).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1318–1322, October, 1994.     

Dynamics of certain class of critically bounded entire transcendental functions

Let E denote the class of all transcendental entire functions for z∈C and an?0 for all n?0 such that f(x)>0 for x<0 and the set of all (finite) singular values of f forms a bounded subset of R. For each f∈E, one parameter family is considered. In this paper, we mainly study the dynamics of functions in the one parameter family S. If f(0)≠0, we show that there exists a positive real number λ^∗ (depending on f) such that the bifurcation and the chaotic burst occur in the dynamics of functions in the one parameter family S at the parameter value λ=λ^∗. If f(0)=0, it is proved that the Julia set of fλ is equal to the complement of the basin of attraction of the super attracting fixed point 0 for all λ>0. It is also shown that the Fatou set F(fλ) of fλ is connected whenever it is an attracting basin and the immediate basin contains all the finite singular values of fλ. Finally, a number of interesting examples of entire transcendental functions from the class E are discussed.     

On the Fatou components of two permutable transcendental entire functions

Let f and g be two distinct permutable transcendental entire functions. Suppose further that q(g)=aq(f)+b for some nonconstant polynomial q(z) and constants a(≠0), . In this article, we will investigate the dynamical properties of f and g and show that they have the same Fatou sets with the same components.     

On the best polynomial approximation of entire transcendental functions of generalized order

We prove a Hadamard-type theorem that associates the generalized order of growth of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.     

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.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

Bounded Fatou components of transcendental entire functions with order less than 1/2

Let f be a transcendental entire function with order ρ 12and let σ be a sufficiently large constant. We prove that if there exists r0 1 such that, for all r r0 and any small ε 0,M(rσ, f) ≥ M(r, f)σ+ε,then every component of the Fatou set F(f) is bounded.     

On the dynamics of a family of entire functions

We prove that for any bounded type irrational number 0θ1,the boundary of the Siegel disk of fα(z)=e2πiθsin(z)+αsin3(z),α∈C,which centered at the origin,is a quasicircle passing through 2,4 or 6 critical points of fαcounted with multiplicity.     

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.