Herman Ring Research Articles

On Fatou Sets Containing Baker Omitted Value

An omitted value of a transcendental meromorphic function f is called a Baker omitted value, in short bov if there is a disk D centered at the bov such that each component of the boundary of \(f^{-1}(D)\) is bounded. Assuming that the bov is in the Fatou set of f, this article investigates the dynamics of the function. Firstly, the connectivity of all the Fatou components are determined. If U is the Fatou component containing the bov then it is proved that a Fatou component \(U'\) is infinitely connected if and only if it lands on U, i.e. \(f^{k}(U') \subset U\) for some \(k \ge 1\). Every other Fatou component is either simply connected or lands on a Herman ring. Further, assuming that the number of critical points in the Fatou set whose forward orbits do not intersect U is finite, we have shown that the connectivity of each Fatou component belongs to a finite set. This set is independent of the Fatou components. It is proved that the Fatou component containing the bov is completely invariant whenever it is forward invariant. Further, if the invariant Fatou component is an attracting domain and compactly contains all the critical values of the function then the Julia set is totally disconnected. Baker domains are shown to be non-existent whenever the bov is in the Fatou set. It is also proved that, if there is a 2-periodic Baker domain (these are not ruled out when the bov is in the Julia set), or a 2-periodic attracting or parabolic domain containing the bov then the function has no Herman ring. Some examples exhibiting different possibilities for the Fatou set are discussed. This includes the first example of a meromorphic function with an omitted value which has two infinitely connected Fatou components.

On periods of Herman rings and relevant poles

Possible periods of Herman rings are studied for general meromorphic functions with at least one omitted value. A pole is called H-relevant for a Herman ring H of such a function f if it is surrounded by some Herman ring of the cycle containing H. In this article, a lower bound on the period p of a Herman ring H is found in terms of the number of H-relevant poles, say h. More precisely, it is shown that \(p\ge \frac{h(h+1)}{2}\) whenever \(f^j(H)\), for some j, surrounds a pole as well as the set of all omitted values of f. It is proved that \(p \ge \frac{h(h+3)}{2}\) in the other situation. Sufficient conditions are found under which equalities hold. It is also proved that if an omitted value is contained in the closure of an invariant or a two periodic Fatou component then the function does not have any Herman ring.

DYNAMICS OF THE FAMILY tan z2

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $\lambda \tan z^2$ for $ \lambda \in \mathbb C^*$. In the dynamical plane, I prove that there is no Herman ring and the Julia set is a Cantor set for the maps when the parameter is in the hyperbolic component containing the origin. Julia set is connected for the maps when the parameters are in other hyperbolic components in the parameter plane.

Herman rings with small periods and omitted values

All possible arrangements of cycles of three periodic as well as four periodic Herman rings of transcendental meromorphic functions having at least one omitted value are determined. It is shown that if p = 3 or 4, then the number of p-cycles of Herman rings is at most one. We have also proved a result about the non-existence of a 3-cycle and a 4-cycle of Herman rings simultaneously. Finally some examples of functions having no Herman ring are discussed.

Surgery on Herman rings of the standard Blaschke family

Let $B_{\alpha ,a}$ be the Blaschke product of the following form:\begin{document}${B_{\alpha ,a}}(z) = {e^{2\pi {\rm{\mathbf{i}}}\alpha }}{z^{d + 1}}{(\frac{{z - a}}{{1 - az}})^d}.$\end{document}If $B_{\alpha ,a}|_{S^1}$ is analytically linearizable, then there is a Herman ring admitting the unit circle as an invariant curve in the dynamical plane of $B_{\alpha ,a}$. Given an irrational number $θ$, the parameters $(\alpha ,a)$ such that $B_{\alpha ,a}|_{S^1}$ has rotation number $θ$ form a curve $T_d(θ)$ in the parameter plane. Using quasiconformal surgery, we prove that if $θ$ is of Brjuno type, the curve can be parameterized real analytically by the modulus of the Herman ring, from $a=M(θ)$ up to $∞$ with $M(θ)≥q 2d+1$, for which the Herman ring vanishes.Moreover, we can show that for a certain set of irrational numbers $θ ∈ \mathcal {B}\setminus\mathcal {H}$, the number $M(θ)$ is strictly greater than $2d+1$ and the boundary of the Herman rings consist of two quasicircles not containing any critical point.

The Fine Structure of Herman Rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.

Rational maps without Herman rings

Let f f be a rational map with degree at least two. We prove that f f has at least two disjoint and infinite critical orbits in the Julia set if it has a Herman ring. This result is sharp in the following sense: there exists a cubic rational map having exactly two critical grand orbits but also having a Herman ring. In particular, f f has no Herman rings if it has at most one infinite critical orbit in the Julia set. These criterions derive some known results about the rational maps without Herman rings.

Herman rings of meromorphic maps with an omitted value

We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period one or two do not exist. Functions with a single pole or with at least two poles one of which is an omitted value have no Herman ring. Every doubly connected periodic Fatou component is a Herman ring.

Classification of invariant Fatou components for dissipative Hénon maps

Fatou components for rational endomorphisms of the Riemann sphere are fully classified and play an important role in our view of one-dimensional dynamics. In higher dimensions, the situation is less satisfactory. In this work we give a nearly complete classification of invariant Fatou components for moderately dissipative Henon maps. Namely, we prove that any such a component is either an attracting or parabolic basin, or the basin of a rotation domain. More specifically, recurrent Fatou components were classified about 20 years ago (modulo the problem of existence of Herman ring basins), while in this paper we prove that non-recurrent invariant Fatou components are semi-parabolic basins. Most of our methods apply in a more general setting.

CONSTRUCTING HERMAN RINGS BY TWISTING ANNULUS HOMEOMORPHISMS

AbstractLet F(z) be a rational map with degree at least three. Suppose that there exists an annulus $H \subset \widehat {\mathbb {C}}$ such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

HERMAN RINGS OF BLASCHKE PRODUCTS OF DEGREE 3

Let Fa,λbe the Blaschke product of the form Fa,λ= λz2((z - a)/(1 - āz)) and α denote an irrational number satisfying the Brjuno condition. Henriksen [1997] showed that for any α there exists a constant a0≧ 3 and a continuous function λ(a) such that Fa,λ(a)possesses an Herman ring and also that modulus M(a) of the Herman ring approaches 0 as a approaches a0. It is remarked that the question whether a0= 3 holds or not is open. According to the idea of Fagella and Geyer [2003] we can show that for a certain set of irrational rotation numbers, a0is strictly larger than 3.

HERMAN RINGS AND ARNOLD DISKS

For $(\l,a)\in \matbb{C}^*\times \mathbb{C}$ , let $f_{\lambda,a}$ be the rational map defined by $f_{\lambda,a}(z) \,{=}\, \lambda z^2 {(az+1)/(z+a)}$ . If $\alpha\in \mathbb{R}/\mathbb{Z}$ is a Brjuno number, we let ${\cal D}_\alpha$ be the set of parameters $(\lambda,a)$ such that $f_{\lambda,a}$ has a fixed Herman ring with rotation number $\alpha$ (we consider that $({\it e}^{2i\pi\alpha}{,}0)\,{\in}\, {\cal D}_\alpha$ ). Results obtained by McMullen and Sullivan imply that, for any $g\in {\cal D}_\alpha$ , the connected component of ${\cal D}_\alpha\cap (\mathbb{C}^*\times(\mathbb{C}\setminus \{0,1\}))$ that contains g is isomorphic to a punctured disk. We show that there is a holomorphic injection $\cal{F}_\alpha\,{:}\,\mathbb{D}\,{\longrightarrow}\, {\cal D}_\alpha$ such that $\cal{F}_\alpha(0) = ({\it e}^{2i\pi \alpha},0)$ and $\cal{F}_\alpha'(0)=(0,r_\alpha),$ where $r_\alpha$ is the conformal radius at 0 of the Siegel disk of the quadratic polynomial $z\longmapsto {\it e}^{2i\pi \alpha}z(1+z)$ . As a consequence, we show that for $a\in (0,1/3)$ , if $f_{\l,a}$ has a fixed Herman ring with rotation number $\alpha$ and if $m_a$ is the modulus of the Herman ring, then, as $a\,{\rightarrow}\,0$ , we have ${\it e}^{\pi m_a} \,{=} ({r_\alpha}/{a}) + {\cal O}(a).$ We finally explain how to adapt the results to the complex standard family $z\,{\longmapsto} \lambda z {\it e}^{({a}/{2})(z-1/z)}$ .

Dynamics of composite functions meromorphic outside a small set

Let M denote the class of functions f meromorphic outside some compact totally disconnected set E = E ( f ) and the cluster set of f at any a ∈ E with respect to E c = C ˆ \\ E is equal to C ˆ . It is known that class M is closed under composition. Let f and g be two functions in class M, we study relationship between dynamics of f ○ g and g ○ f . Denote by F ( f ) and J ( f ) the Fatou and Julia sets of f. Let U be a component of F ( f ○ g ) and V be a component of F ( g ○ f ) which contains g ( U ) . We show that under certain conditions U is a wandering domain if and only if V is a wandering domain; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U unless U is a Siegel disk or Herman ring.

An elementary proof that Herman's Ring is [formula omitted

Herman's Ring [Inform. Process. Lett. 35 (1990) 63; http://www.cs.uiowa.edu/ftp/selfstab/H90.ps.gz] is an algorithm for self-stabilization of N identical processors connected uni-directionally in a synchronous ring; in its original form it has been shown to achieve stabilization, with probability one, in expected steps O ( N 2 log N ) . We give an elementary proof that the original algorithm is in fact O ( N 2 ) ; and for the special case of three tokens initially we give an exact (quadratic) solution of 4 a b c / N , where a , b , c are the tokens' initial separations. Thus the algorithm overall has worst-case expected running time of Θ ( N 2 ) . Although we use only simple matrix algebra in the proof, the approach was suggested by the general notions of abstraction, nondeterminism and probabilistic variants [A. McIver, C. Morgan, Refinement and Proof for Probabilistic Systems, Technical Monographs in Computer Science, Springer-Verlag, New York, 2004]. It is hoped they could also be useful for other, similar problems. We conclude with an open problem concerning the worst-case analysis.

On holomorphic critical quasi-circle maps

The so-called Herman–Światek theorem is generalized to holomorphic self-homeomorphisms of quasi-circles. This result implies an unpublished theorem of Michel Herman: If a Siegel disk or Arnold–Herman ring for a rational map has a boundary component, which is a quasi-circle containing a critical point, then the associated rotation number is Diophantine of exponent 2.

Surgery on Herman rings of the complex standard family

We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67–98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}.

Periodic Fatou Components of Meromorphic Functions

In iteration theory of rational functions, it is well known that any Fatou component is mapped onto another in an n-to-1 manner, and that periodic components are simply, doubly or infinitely connected. For meromorphic functions, the situation is much more complicated. Using Ahlfors' theory of covering surfaces, we prove that Fatou components are mapped ‘nearly’ onto others, and that periodic components are again simply, doubly or infinitely connected. Instead of considering meromorphic functions with only one essential singularity, we allow countable sets of singularities and partly even sets of logarithmic capacity zero. It remains open whether doubly connected periodic components of meromorphic functions with only one singularity are necessarily Herman rings (as holds for rational functions). However, there is a function with two singularities and a doubly connected periodic component which is not an Herman ring. 1991 Mathematics Subject Classification 30D05, 30D30.

Periodic points for actions of tori in Stein manifolds

In one complex variable dynamics, Sullivan’s theorem ([6]) gives a complete classification of the Fatou components that can appear for a rational map f . Consequently we can have only periodic components U and U can be one of the following: 1. U attracting basin; 2. U parabolic domain; 3. U Siegel disk; 4. U Herman ring. Cases 3. and 4. are called rotation domains. In these cases, the rational map f is conjugated on U to an irrational rotation, hence by taking all the iterates of f and their limits we obtain an S1-action on U which has at most one fixed (periodic) point in U . The goal of this paper is to give conditions when the generalization of the one variable situation is true. In particular we consider actions of tori on Stein manifolds and study their periodic (fixed) points.

Dynamics of the Complex Standard Family

The complexification of the standard family of circle maps θ↦θ+α+βsin(θ)mod(2π), whose parameter space contains the well-known Arnold tongues, is given byFαβ(ω)=ωeiαe(β/2)(ω−1/ω), a holomorphic map of C* with essential singularities at 0 and ∞. For real values of the parameters, we study the dynamical plane of the familyFαβ. Near the essential singularities we prove the existence ofhairsin the Julia set, an invariant set of curves organized by some symbolic dynamics, whose points (that are not endpoints) tend exponentially fast to 0 or ∞ under iteration. For β<1, we give a complex interpretation of the bifurcations of the family of circle maps. More precisely, we give a new characterization of the rational Arnold tongues in terms of some of the hairs attaching to the unit circle. For certain irrational rotation numbers, we show that the Fatou set consists exclusively of a Herman ring and its preimages. For β>1 we prove that, under certain conditions, all hairs end up attached to the unit circle as we increase the parameter.

Analytic self-maps of the punctured plane

Let f be a non-Möbius self-map of C *. Write J *(f) for the Julia Set of f in Ĉ and J(f) for the closure of J *(f) in Č. Then we have one of the three cases (i)J(f), J *(f) are connected, (ii)J(f) is connected, J *(f) has infinitely many components, (iii)J(f) has two components, J *(f) has infinitely many components. All three cases can occur. It is known that F(f) has at most one multiply-connected domain A whose connectivity must in fact be two. If in addition A is relatively compact in nC *, then either (i) A is a Herman ring, (ii) A is pre-periodic but nor periodic, or (iii) A is a wandering component Example of all three cases are constructed.