David extension of circle homeomorphisms, welding, mating, and removability (original) (raw)

On conjugacy of homeomorphisms of the circle possessing periodic points

Journal of Mathematical Analysis and Applications, 2007

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism F : S 1 → S 1 has a periodic point of period n iff there exist a positive integer q < n relatively prime to n and a homeomorphism Φ : S 1 → S 1 such that the lift of Φ −1 • F • Φ restricted to [0, 1] has the form

Circle packings, kissing reflection groups and critically fixed anti-rational maps

Forum of Mathematics, Sigma

In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.

On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections

Conformal Geometry and Dynamics of the American Mathematical Society

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}_g is (non-quasiconformally) homeomorphic to Λ H \Lambda _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H \Lambda _H and J g \mathcal {J}_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H \Lambda _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H \Lambda _H is the classical Apollonian gasket), we give a quasiregular model for the above actions whi...

On conformal welding homeomorphisms associated to Jordan curves

Annales Academiae Scientiarum Fennicae Series A I Mathematica, 1990

tr'or any Jordan curve C on the Riemann sphere, the conformal welding homeomorphism, Weld(C), of the circle or real line onto itself is obtained by comparing the bLundary values of the Riemann mappings of the unit disk or upper half-plane onto the two complementa,ry Jordan regions separated by C. In Part I we show that a family of infinitely spiralling curves lie intermediate between smooth curves and curves with corners, in the sense that the welding for the spirals has a non-differentiable but Lipschitz singularity. We also show how the behaviour of Riemann mapping germs under analytic continuation implies that spirals were the appropriate curves to produce such weldings. A formula relating explicitly the rate of spiralling to the extent of non-differentiablity is proved by two methods, and related to known theorems for chord-arc curves. Part II studies families of Jordan curves possessing the same welding. We utilise curves with positive area and the Ahlfors-Bers generalised Riemann mapping theorem to build an infinite dimensional "Teichmiiller space" of curves all having the same welding homeomorphism. A point of view on welding as a prob-Iem of extension of conformal structure is also derived in this part.

On a factorization of homeomorphisms of the circle possessing periodic points

Journal of Mathematical Analysis and Applications, 2008

We prove that for every orientation-preserving homeomorphism F : S 1 → S 1 possessing periodic points of order n there exist a homeomorphism T : S 1 → S 1 such that T n = id and a homeomorphism G : S 1 → S 1 without periodic points except fixed points such that

Spinning deformations of rational maps

2004

We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as spinning. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges to infinity in the moduli space of rational maps of a fixed degree. When the limit exists, it has either a parabolic fixed point, or a pre-periodic critical point in the Julia set, depending on the combinatorics of the data defining the deformation. The proofs are soft and rely on two ingredients: the construction of a Riemann surface containing the closure of the family, and an analysis of the geometric limits of some simple dynamical systems. An interpretation in terms of Teichmüller theory is presented as well. 52 SPINNING DEFORMATIONS OF RATIONAL MAPS 53 54 KEVIN M. PILGRIM AND TAN LEI

Polynomial diffeomorphisms of C2: VII. Hyperbolicity and external rays

Annales Scientifiques de l’École Normale Supérieure, 1999

For a polynomial automorphism we study the topology of J, the analogue of the Julia set, in the case where J is connected and hyperbolic. 0 Elsevier, Paris RBSUML -A partir dun automorphisme polynomial de C?, nous ttudions l'analogue de l'ensemble de Julia, note J, dam le cas oh J est hyperbolique et connexe. 0 Elsevier, Paris * Research supported in part by the NSF. ANNALESSCIENTIFIQUESDEL.%COLENORMALESUP&IEURE. -0012-9593/99/04/O Elsevier,Paris 456 E. BEDFORD AND J. SMILLIE

On deformation space analogies between Kleinian reflection groups and antiholomorphic rational maps

Geometric and Functional Analysis

In a previous paper [LLM20], we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces. Contents 1. Introduction 1 2. Degeneration of anti-Blaschke products 8 3. Realization of (d + 1)-ended ribbon trees 19 4. Boundedness and mutual interaction of deformation spaces 29 5. Markov partitions and monodromy representations 39 Appendix A. Laminations, automorphisms and accesses 42 Appendix B. Shared matings, self-bumps and disconnected roots 47 References 52