Thurston Maps Research Articles

CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

Cannon–Thurston maps for hyperbolic free group extensions

This paper gives a detailed analysis of the Cannon{Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of nite rank at least 3 and consider a convex cocompact subgroup Out(F), i.e. one for which the orbit map from into the free factor complex of F is a quasi-isometric embedding. The subgroup determines an extension E of F, and the main theorem of Dowdall{Taylor (DT1) states that in this situation E is hyperbolic if and only if is purely atoroidal. Here, we give an explicit geometric description of the Cannon{Thurston maps @F! @E for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon{Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig (KL5) which treats the special case where is innite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of to the space of laminations of the free group (with the Chabauty topology) is not continuous.

Veering triangulations and Cannon–Thurston maps

Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon–Thurston and Bowditch. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.

Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

The key result in the present paper is a direct analogue of the celebrated Thurston’s Theorem Douady and Hubbard (Acta Math 171:263–297, 1993) for marked Thurston maps with parabolic orbifolds. Combining this result with previously developed techniques, we prove that every Thurston map can be constructively geometrized in a canonical fashion. As a consequence, we give a partial resolution of the general problem of decidability of Thurston equivalence of two postcritically finite branched covers of $$S^2$$ (cf. Bonnot et al. Moscow Math J 12:747–763, 2012).

Weak expansion properties and large deviation principles for expanding Thurston maps

In this paper, we prove that an expanding Thurston map f:S2→S2 is asymptotically h-expansive if and only if it has no periodic critical points, and that no expanding Thurston map is h-expansive. As a consequence, for each expanding Thurston map without periodic critical points and each real-valued continuous potential on S2, there exists at least one equilibrium state. For such maps, we also establish large deviation principles for iterated preimages and periodic points. It follows that iterated preimages and periodic points are equidistributed with respect to the unique equilibrium state for an expanding Thurston map without periodic critical points and a potential that is Hölder continuous with respect to a visual metric on S2.

Measurable rigidity for Kleinian groups

Let$G,H$be two Kleinian groups with homeomorphic quotients$\mathbb{H}^{3}/G$and$\mathbb{H}^{3}/H$. We assume that$G$is of divergence type, and consider the Patterson–Sullivan measures of$G$and$H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map$\widehat{k}$from the limit set$\unicode[STIX]{x1D6EC}_{G}$of$G$to that of$H$is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such$\widehat{k}$always exists. In fact, we shall construct$\widehat{k}$concretely from the Cannon–Thurston maps of$G$and$H$.

A decomposition theorem for Herman maps

In 1980s, Thurston established a topological characterization theorem for postcritically finite rational maps. In this paper, a decomposition theorem for a class of postcritically infinite branched covering termed Herman map is developed. It's shown that every Herman map can be decomposed along a stable multicurve into finitely many Siegel maps and Thurston maps, such that the combinations and rational realizations of these resulting maps essentially dominate the original one. This result is motivated by a non-expanding version of McMullen's problem, and Thurston's theory on characterization of rational maps. It enables us to prove a Thurston-type theorem for rational maps with Herman rings.

Ending Laminations and Cannon–Thurston Maps

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.

On Cannon–Thurston maps for relatively hyperbolic groups

Abstract.Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).

Limits of limit sets I

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon–Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometric limit is geometrically finite, the maps on limit sets converge pointwise but not uniformly.

Nearly Euclidean Thurston maps

We take an in-depth look at Thurston’s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S 2 → S 2 f\colon S^2\to S^2 whose local degree at every critical point is 2 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics.

An algebraic characterization of expanding Thurston maps

Let $f\colon S^2 \to S^2$ be a postcritically finite branched covering mapwithout periodic branch points. We give necessary and sufficientalgebraic conditions for $f$ to be homotopic, relative to itspostcritical set, to an expanding map $g$.

Cannon–Thurston maps for pared manifolds of bounded geometry

Let N^h be a hyperbolic 3-manifold of bounded geometry corresponding to a hyperbolic structure on a pared manifold (M,P). Further, suppose that (\partial{M} - P) is incompressible, i.e. the boundary of M is incompressible away from cusps. Further, suppose that M_{gf} is a geometrically finite hyperbolic structure on (M,P). Then there is a Cannon- Thurston map from the limit set of M_{gf} to that of N^h. Further, the limit set of N^h is locally connected. This answers in part a question attributed to Thurston.

Exponential Thurston maps and limits of quadratic differentials

We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers g : C → C ∖ { 0 } g\colon \mathbb {C}\to \mathbb {C}\setminus \{0\} such that 0 0 has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map λ e z \lambda e^z or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston’s topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree. One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.

Analytic models of pseudo-Anosov maps

AbstractWe give a new proof of the existence of analytic models of pseudo-Anosov maps. The persistence properties of Thurston's maps ensure that any Co-perturbation of them presents all their dynamical features. Using Lyapunov functions of two variables we are able to choose certain analytic perturbations which do not add any new dynamical behaviour to the original pseudo-Anosov map.