Hee Oh, Nimish Shah
For a circle packing P on the sphere invariant under a geometrically finite Kleinian group, we compute the asymptotic of the number of circles in P of spherical curvature at most $T$ which are contained in any given region.
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Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either the critical exponent of Gamma is strictly bigger than 1 or P does not contain an infinite bouquet of tangent circles glued at a parabolic fixed point of Gamma. Our construction also works for P invariant under a geometrically infinite group Gamma, provided Gamma admits a finite…
Consider a general circle packing $\mathcal{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $Γ$. When $Γ$ is convex-cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathcal{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah. In view of the recent result of McMullen-Mohammadi-Oh, our effective circle counting theorem applies to the circles contained in the limit set of a convex-cocompact but non-cocompact Kleinian group whos…
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schottky dances, etc.
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 $H$ whose limit set is an Apollonian-like gasket $Λ_H$. We design a surgery that relates $H$ to a rational map $g$ whose Julia set $\mathcal{J}_g$ is (non-quasiconformally) homeomorphic to $Λ_H$. We show for a large class of triangulations, however, the groups of quasisymmetries of $Λ_H$ and $\mathcal{J}_g$ are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case …
Thurston and Calegari-Dunfield showed that the fundamental group of some tautly foliated hyperbolic 3-manifold acts on the circle in a distinctive way that the action preserves some structure of S^1, so-called a circle lamination. Indeed, a large class of Kleinian groups acts on the circle preserving a circle lamination. In this paper, we are concerned with the converse problem that a group acting on the circle with at least two invariant circle laminations is Kleinian. We prove that a subgroup of the orientation preserving circle homeomorphism group is a Fuchsian group whose quotient orbifold…
Rigidity of Kleinian groups via self-joinings
本文通过引入自连接高阶动力学的新视角，证明了在极限集上具有共形边界映射的非Fuchsian Kleinian群的新刚性定理，并回答了McMullen关于零体积四面体顶点映射刚性的问题。
Circle packings, renormalizations and subdivision rules
本文利用 Teichmüller 空间上剥皮映射的迭代，为满足特定细分规则的圆填充发展了重整化理论，证明重整化算子一致收缩，从而完整解决了此类圆填充的存在性与模问题，并证明了其几何不灵活性与组合刚性。
Combining Rational maps and Kleinian groups via orbit equivalence
本研究提出复多项式与Kleinian曲面群全纯配对的轨道等价框架，定义高阶Bowen-Series映射，明确配对Teichmüller空间分支，证明圆周Fuchsian群无轨道等价刚性，并分类可配对的Kleinian Bers边界群。
A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.
For certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit's deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of quadratic maps, and of matings between the quadratic map $z\to z^2$ and circle-packing representations of the free product $C\_2*C\_3$ of cyclic groups of order 2 and 3.
Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its limit set and Omega(G) = S \ Lambda(G) (S = the sphere) its set of discontinuity. Let delta(G) be the critical exponent for the Poincaré series and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply connected component of Omega(G). We prove that (1) delta(G) = dim(Lambda_c). (2) A simply connected component Omega is either a disk or dim(Omega)>1$. (3) Lambda(G) is either totally disconnected, a circle or has dimension > 1, (4) G is geometrically infinite iff dim(Lambda)=2. (5) If G_n \to G algebrai…
Torus counting and self-joinings of Kleinian groups
该论文针对Kleinian群自连接下的一类d维环面堆积，建立了小体积环面的计数与等分布渐近公式，推广了Oh-Shah的圆堆积结果。
Rigidity of Kleinian groups via self-joinings: measure theoretic criterion
本文通过自联结子群的遍历理论与共形测度理论，证明了扎里斯基稠密凸紧克莱因群表示的测度论刚性二分法，并将结果推广至发散型子群。
This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of Möbius transformations on the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$), including the classical Apollonian gasket and some round Sierpiński carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution…
We give conditions for the existence of the Minkowski content of limit sets stemming from infinite conformal graph directed systems. As an application we obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae of the Minkowski content, and prove the analytic dependence on the initial circles. Further, we are able to link the fractal Euler characteristic, as well as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour of the circle counting function studied by Kontorovich and Oh. These results lead to a new interpretation and an alternative formula …
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