arXiv research

Study shows Julia sets and gasket limit sets are quasiconformally different.

Find limiting sets for digital cones and suspensions.

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 geometr…

Study shows non-symmetric convex sets have full boundary limits.

Rare Teichmüller disks converge to small limit sets.

The paper characterizes subgroup stability via limit sets on the Morse boundary.

Study continuity of limit sets in symmetric spaces.

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove…

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) th…

We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being …

Study calculates Kulkarni limit sets for quaternionic projective groups.

Smooth knots in complex hyperbolic plane limit sets to chains or R-circles.

The paper connects geodesic flows and limit sets on visibility manifolds.

Study contractibility of boundaries in convex sets and limit sets of subgroups.

Study shows spectral gaps limit points on surfaces.

This note fixes a small gap in Kerckhoff's proof that the limit set of the handlebody set has measure zero.

We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

For a torsion free Kleinian group $Γ$ without parabolics, we consider the decomposition of the limit set $L(Γ)$ into conical and ending limit sets and compare the Patterson-Sullivan measure with the harmonic measure on $L(Γ)$ when $L(Γ)=S^2_\infty$.

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

Two groups with specific limit sets in hyperbolic spaces are identified.

We relate the L^2 cohomology of a complete hyperbolic manifold to the invariant currents on its limit set.

In this paper we prove that the limit set of any Weil-Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil-Petersson geodesics with minimal nonuniquely ergodic ending laminations and limit…

Paper studies Hausdorff dimensions of specific limit sets for groups on curved spaces.

Constructs hyperbolic reflection groups with 3D limit sets.

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respe…

We characterise completely when limit sets, as parametrised by Cannon-Thurston maps, move discontinuously for a sequence of algebraically convergent quasi-Fuchsian groups.

The fundamental group of a hyperbolic manifold acts on the limit set, giving rise to a cross-product C^* algebra. We construct nontrivial K-cycles for the cross-product algebra, thereby extending some results of Connes and Sullivan to higher dimensions. We also show how the Patterson-Sullivan measure on the limit set c…

In this paper, we prove a limit set intersection theorem in relatively hyperbolic groups. Our approach is based on a study of dynamical quasiconvexity of relatively quasiconvex subgroups. Using dynamical quasiconvexity, many well-known results on limit sets of geometrically finite Kleinian groups are derived in general…

We construct an example of a Teichmueller geodesic ray whose limit set in Thurston boundary of Teichmueller space is a d-dimensional simplex.

Study on mapping class groups of non-orientable surfaces, proving some conjectures and refuting others.

Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $Λ_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $Λ_H$ consisting of the points to wh…

While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups $Γ$ of arithmetic groups in $PSL(2,C)^q \times PSL(2,R)^r$ with $q+r>1$ and the…

Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.

We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension $<1$ are free. On the other hand we construct for any $ε>0$ examples of non-free purely hyperbolic Kleinian groups whose limit set is a Cantor set of Hausdorff dimension $<1+ε$.

Anosov groups' measures on limit sets are uniquely determined by their dimension.

New framework captures non-autonomous IFS limit set topology.

In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not a…

Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in 'most' cases. In particular, we show that it holds when the domain of discontinui…

New examples show non-integer Hausdorff dimensions in collapsing spaces.

In this paper we give a description of the possible limit sets of finitely generated subgroups of irreducible lattices in $PSL(2,R)^r$.

In this paper we construct examples of Weil-Petersson geodesics with nonminimal ending laminations which have 1-dimensional limit sets in the Thurston compactification of Teichmüller space.

Study shows continuity and geometric regularity of Kähler-Ricci flow blow-up limits.

We explore the limit set of a particular spherical CR uniformization of a cusped hyperbolic manifold. We prove that the limit set is the closure of a countable union of $\mathbb{R}$-circles, is connected, and contains a Hopf link with three components; we also show that the fundamental group of its complement in $S^3$ …

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where eve…

Defines and extends Lie algebroid prolongations in convenient settings.

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

The Basilica Julia set is universally equivalent to other complex dynamics sets.