Related papers: Hyperbolic-parabolic deformations of rational maps

We extend Thurston's combinatorial criterion for postcritically finite rational maps to a class of rational maps with bounded type Siegel disks. The combinatorial characterization of this class of Siegel rational maps plays a special role…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map…

We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in…

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of…

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties…

We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image…

Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let $v$ be a vertex of $T$. Let $({X_v},d_v)$ denote the hyperbolic metric space corresponding to $v$. Then $i : X_v \rightarrow X$…

In this paper, we introduce cosine Thurston maps. In particular, we construct postsingularly finite topological cosine maps and focus on such maps with strictly preperiodic critical points. We use the techniques of Hubbard, Schleicher, and…

We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.

Topological mating is an combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map $F$ on the…

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on…

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary -- cornered asymptotically hyperbolic manifolds -- and proves a theorem of Cartan-Hadamard type near infinity for the…

Thurston related $\mathbb{C}{\rm P}^1$-structures (complex projective structures) and equivariant pleated surfaces in the hyperbolic-three space $\mathbb{H}^3$, in order to give a parameterization of the deformation space of $\mathbb{C}{\rm…

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…

Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…

Let $f:(\mathbb{P}^1,P)\to(\mathbb{P}^1,P)$ be a postcritically finite rational map with postcritical set $P$. William Thurston showed that $f$ induces a holomorphic pullback map $\sigma_f:\mathcal{T}_P\to\mathcal{T}_P$ on the Teichm\"uller…

Given two hyperbolic surfaces and a homotopy class of maps between them, Thurston proved that there always exists a representative minimizing the Lipschitz constant. While not unique, these minimizers are rigid along a geodesic lamination.…

In this paper we present local Sternberg conjugation theorems near attracting fixed points for lattice systems. The interactions are spatially decaying and are not restricted to finite distance. The conjugations obtained retain the same…

A Thurston map is a branched covering map from $\S^2$ to $\S^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on…