Topological characterisation of rational maps with Siegel disks

We 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.

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.

Renormalization in the golden-mean semi-Siegel Hénon family: universality and non-rigidity

It was recently shown in Gaidashev and Yampolsky [Golden mean Siegel disk universality and renormalization. Preprint, 2016, arXiv:1604.00717] that appropriately defined renormalizations of a sufficiently dissipative golden-mean semi-Siegel Hénon map converge super-exponentially fast to a one-dimensional renormalization fixed point. In this paper, we show that the asymptotic two-dimensional form of these renormalizations is universal and is parameterized by the average Jacobian. This is similar to the limit behavior of period-doubling renormalizations in the Hénon family considered in de Carvalho et al [Renormalization in the Hénon family, I: universality but non-rigidity. J. Stat. Phys.121 (5/6) (2006), 611–669]. As an application of our result, we prove that the boundary of the golden-mean Siegel disk of a dissipative Hénon map is non-smoothly rigid.

Singular values and bounded Siegel disks

Let f be an entire transcendental function of finite order and Δ be a forward invariant bounded Siegel disk for f with rotation number in Herman's class $\mathcal{H}$. We show that if f has two singular values with bounded orbit, then the boundary of Δ contains a critical point. We also give a criterion under which the critical point in question is recurrent. We actually prove a more general theorem with less restrictive hypotheses, from which these results follow.

On PZ type Siegel disks of the sine family

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.

CONSEQUENCES OF THE FUNDAMENTAL CONJECTURE FOR THE MOTION ON THE SIEGEL–JACOBI DISK

We find the homogeneous Kähler diffeomorphism FC which expresses the Kähler two-form on the Siegel–Jacobi domain [Formula: see text] as the sum of the Kähler two-form on ℂ and the one on the Siegel ball [Formula: see text]. The classical motion and quantum evolution on [Formula: see text] determined by a linear Hamiltonian in the generators of the Jacobi group [Formula: see text] is described by a Riccati equation on [Formula: see text] and a linear first-order differential equation in z ∈ ℂ, where H1 denotes the three-dimensional Heisenberg group. When the transformation FC is applied, the first-order differential equation for the variable z ∈ ℂ decouples of the motion on the Siegel disk. Similar considerations are presented for the Siegel–Jacobi space [Formula: see text], where [Formula: see text] denotes the Siegel upper half-plane.

Bounded-type Siegel disks of a one-dimensional family of entire functions

Let 0<θ<1 be an irrational number of bounded type. We prove that for any map in the family (e2πiθz+αz2)ez, α∈ℂ, the boundary of the Siegel disk, with fixed point at the origin, is a quasi-circle passing through one or both of the critical points.