scholarly journals Mating the Basilica with a Siegel disk

2015 ◽  
Vol 19 (12) ◽  
pp. 258-297
Author(s):  
Jonguk Yang
Keyword(s):  
Siegel Disk

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

2009 ◽  
Vol 29 (1) ◽  
pp. 137-164 ◽  
Author(s):  
LINDA KEEN ◽  
GAOFEI ZHANG
Keyword(s):  
Fixed Point ◽  
Critical Points ◽  
Entire Functions ◽  
Irrational Number ◽  
Siegel Disk ◽  
One Dimensional ◽  
The Family

AbstractLet 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.

scholarly journals How regular can the boundary of a quadratic Siegel disk be?

2007 ◽  
Vol 135 (04) ◽  
pp. 1073-1073 ◽  
Author(s):  
Xavier Buff ◽  
Arnaud Chéritat
Keyword(s):  
Siegel Disk

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

2012 ◽  
Vol 10 (01) ◽  
pp. 1250076 ◽  
Author(s):  
STEFAN BERCEANU
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Three Dimensional ◽  
Quantum Evolution ◽  
Siegel Disk ◽  
Classical Motion ◽  
First Order ◽  
Jacobi Group ◽  
Upper Half Plane ◽  
The One

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.

On PZ type Siegel disks of the sine family

2014 ◽  
Vol 36 (3) ◽  
pp. 973-1006
Author(s):  
GAOFEI ZHANG
Keyword(s):  
Jordan Curve ◽  
Critical Points ◽  
Siegel Disk ◽  
Sine Family ◽  
Rotation Numbers

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 $-{\it\pi}/2$.

scholarly journals Singular values and bounded Siegel disks

2017 ◽  
Vol 165 (2) ◽  
pp. 249-265
Author(s):  
ANNA MIRIAM BENINI ◽  
NÚRIA FAGELLA
Keyword(s):  
Critical Point ◽  
Finite Order ◽  
Rotation Number ◽  
General Theorem ◽  
Singular Values ◽  
Transcendental Function ◽  
Siegel Disk ◽  
Entire Transcendental Function

AbstractLet 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.

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