Siegel Disks Research Articles

On possible limit functions on a Fatou component in non-autonomous iteration

Abstract The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

The high type quadratic Siegel disks are Jordan domains

Let \alpha be an irrational number of sufficiently high type and suppose P_{\alpha}(z)=e^{2\pi\textup{i}\alpha}z+z^{2} has a Siegel disk \Delta_{\alpha} centered at the origin. We prove that the boundary of \Delta_{\alpha} is a Jordan curve, and that it contains the critical point -e^{2\pi\textup{i}\alpha}/2 if and only if \alpha is a Herman number.

Siegel disks on rational surfaces

We show the existence of a rational surface automorphism of positive entropy with a given number of Siegel disks. Moreover, among automorphisms obtained from quadratic birational maps on the projective plane fixing irreducible cubic curves, we find out an automorphism of positive entropy with multiple Siegel disks.

The long-term behavior of the p-adic Sigmoid Beverton-Holt dynamical systems in the projective line [formula omitted

In this paper we study the long-term behavior of the p-adic dynamical systems associated with the Sigmoid Beverton-Holt model that arises in population dynamics. The corresponding fixed points, maximal Siegel disks, attractors, and periodic trajectories are analyzed in the projective line P1(Qp). Moreover, the Julia and Fatou sets associated with these dynamical systems are also examined.

Wandering domains arising from Lavaurs maps with siegel disks

The classification of Fatou components for rational functions was concluded with Sullivan's proof of the No Wandering Domains Theorem in 1985. In 2016 it was shown, in joint work of the first and last author with Buff, Dujardin and Raissy, that wandering domains do exist in higher dimensions. In fact, wandering domains arise even for a seemingly simple class of maps: polynomial skew products. While the construction gives an infinite dimensional class of examples, and has been extended to polynomial automorphisms of $\mathbb C^4$ by Hahn and the last author, the currently known wandering domains are essentially unique. Our goal in this paper is to construct a second example, arising from similar techniques, but with distinctly different dynamical behavior. Instead of wandering domains arising from a Lavaurs map with an attracting fixed point, we construct a domain arising from a Lavaurs map with a fixed point of Siegel type. Siegel disks are not robust under perturbations, as opposed to attracting fixed points. We prove a necessary and sufficient condition for the existence of a so-called trapping domain for non-autonomous dynamical systems given by sequences of maps converging parabolically towards a Siegel type limit map. Guaranteeing that this condition is satisfied in our current construction requires a reconsideration of the proof of the original wandering domain, as more precise estimates on the rate of convergence towards the Lavaurs map are required. By adapting ideas introduced recently by Bedford, Smillie and Ueda, and by proving the existence of parabolic curves, with control on their domains of definition, we prove that the convergence rate is parabolic.

K3 surfaces, Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.

Ergodicity and Periodic Orbits of $$p$$-Adic $$(1,2)$$-Rational Dynamical Systems with Two Fixed Points

We consider $$(1,2)$$ -rational functions given on the field of $$p$$ -adic numbers $${\mathbb Q}_p$$ . In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then $$(1,2)$$ -rational function is conjugate to a two-parametric $$(1,2)$$ -rational function. Depending on these two parameters we determine type of the fixed points, find Siegel disks and the basin of attraction of the fixed points. Moreover, we classify invariant sets and study ergodicity properties of the function on each invariant set. We describe 2- and 3-periodic orbits of the $$p$$ -adic dynamical systems generated by the two-parametric $$(1,2)$$ -rational functions.

Single and double toral band Fatou components in meromorphic dynamics

We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.

Some transcendental entire functions with irrationally indifferent fixed points

Let $S$ be the set of all transcendental entire functions of the form $P(z) \exp (Q(z))$, where $P$ and $Q$ are polynomials. In this paper, by using the theory of polynomial-like mappings, we construct various kinds of functions in $S$ with irrationally indifferent fixed points as follows: (1) We construct functions in $S$ with bounded type Siegel disks centered at points other than the origin bounded by quasicircles containing critical points. This is an extension of Zakeri's result in [24] for $f \in S$. (2) We construct functions in $S$ with Cremer points whose multipliers satisfy some Cremer's condition in [6] only for rational functions. Our method shows that this condition can be applicable even in some transcendental cases. (3) For any integer $d \geq 2$ and some $c \in {\mathbf C} \setminus \{0\}$, we show that the function of the form $e^{2\pi i \theta}z(1 + cz)^{d-1}e^z\,(\theta \in {\mathbf R} \setminus {\mathbf Q})$ has a Siegel point at the origin if and only if $\theta$ is a Brjuno number. This is an extension of Geyer's result in [11]. (4) For the function of the form $(e^{2\pi i\theta}z+\alpha z^2)e^z \,(\theta \in {\mathbf R} \setminus {\mathbf Q}, \alpha \in {\mathbf C} \setminus \{0\})$, we show that if $\alpha$ and $\theta$ satisfy some condition, then the Siegel disk centered at the origin is bounded by a Jordan curve containing a critical point, which is not a quasicircle. Moreover, we can choose $\alpha$ and $\theta$ so that the Lebesgue measure of the Julia set is positive and can also choose them so that it is zero. This is an extension of Keen and Zhang's result in [13].

Best possibility of the Fatou-Shishikura inequality for transcendental entire functions in the Speiser class

The Speiser class S S is the set of all entire functions with finitely many singular values. Let S q ⊂ S S_q\subset S be the set of all transcendental entire functions with exactly q q distinct singular values. The Fatou-Shishikura inequality for f ∈ S q f\in S_q gives an upper bound q q of the sum of the numbers of its Cremer cycles and its cycles of immediate attractive basins, parabolic basins, and Siegel disks. In this paper, we show that the inequality for f ∈ S q f\in S_q is best possible in the following sense: For any combination of the numbers of these cycles which satisfies the inequality, some T ∈ S q T\in S_q realizes it. In our construction, T T is a structurally finite transcendental entire function.

Quadratic Rational Maps with a 2-Cycle of Siegel Disks

For the family of quadratic rational functions having a $2$-cycle of bounded type Siegel disks, we prove that each of the boundaries of these Siegel disks contains at most one critical point. In the parameter plane, we prove that the locus for which the boundaries of the $2$-cycle of Siegel disks contain two critical points is a Jordan curve.

Hypergeometric groups and dynamics on K3 surfaces

A hypergeometric group is a matrix group modeled on the monodromy group of a generalized hypergeometric differential equation. This article presents a fruitful interaction between the theory of hypergeometric groups and dynamics on K3 surfaces by showing that a certain class of hypergeometric groups and related lattices lead to a lot of K3 surface automorphisms of positive entropy, especially such automorphisms with Siegel disks.

Golden Mean Siegel Disk Universality and Renormalization

We provide a computer-assisted proof of one of the central open questions in one-dimensional renormalization theory -- universality of the golden-mean Siegel disks. We further show that for every function in the stable manifold of the golden-mean renormalization fixed point the boundary of the Siegel disk is a quasicircle which coincides with the closure of the critical orbit, and that the dynamics on the boundary of the Siegel disk is rigid. Furthermore, we extend the renormalization from one-dimensional analytic maps with a golden-mean Siegel disk to two-dimensional dissipative H\'enon-like maps and show that the renormalization hyperbolicity result still holds in this setting.

Bounded critical Fatou components are Jordan domains for polynomials

We prove that any bounded Fatou component of a polynomial of degree at least two, which is not (eventually) a Siegel disk, is a Jordan domain.

On the formulas of meromorphic functions with periodic Herman rings

We construct some explicit formulas of rational maps and transcendental meromorphic functions having Herman rings of period strictly larger than one. This gives an answer to a question raised by Shishikura in the 1980s. Moreover, the formulas of some rational maps with nested Herman rings are also found. To obtain the formulas of transcendental meromorphic functions having periodic Herman rings, a crucial step is to find an explicit family of transcendental entire functions having bounded Siegel disks of all possible periods and rotation numbers. This is based on proving the existence of a Mandelbrot-like set of period one in the corresponding parameter space.

Siegel disks of the tangent family

We study Siegel disks in the dynamics of functions from the tangent family. In particular, we prove that a Siegel disk is unbounded if and only if the boundary of its image contains at least one asymptotic value. Moreover, by using quasiconformal surgery we also construct functions in the above family with bounded Siegel disks.

Location of Siegel capture polynomials in parameter spaces

A cubic polynomial with a marked fixed point 0 is called an IS-capture polynomial if it has a Siegel disk D around 0 and if D contains an eventual image of a critical point. We show that any IS-capture polynomial is on the boundary of a unique bounded hyperbolic component of the polynomial parameter space determined by the rational lamination of the map and relate IS-capture polynomials to the cubic principal hyperbolic domain and its closure.

Topological characterisation of rational maps with Siegel disks

AbstractWe extend Thurston’s topological characterisation theorem for postcritically finite rational maps to a class of rational maps which have a fixed bounded type Siegel disk. This makes a small step towards generalizing Thurston’s theorem to geometrically infinite rational maps.

Renormalization and Siegel disks for complex Hénon maps

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.

The Siegel-Klein Disk: Hilbert Geometry of the Siegel Disk Domain.

We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.