Results 1  10
of
23
Hausdorff dimension and conformal dynamics III: Computation of dimension
 Amer. J. Math
"... This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Di ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials fc(z) = z 2 + c, c ∈ [−1, 1/2]; and (c) the family of rational maps ft(z) = z/t+ 1/z, t ∈ (0, 1]. We also calculate H. dim(Λ) ≈ 1.305688 for the Apollonian gasket, and H. dim(J(f)) ≈ 1.3934 for Douady’s rabbit, where f(z) = z 2 + c
A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups, Ergodic theory and dynamical systems 24
, 2004
"... ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these gro ..."
Abstract

Cited by 22 (12 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates ’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ’strong phase transitions’. 1.
The hyperbolic lattice point count in infinite volume with applications to sieves
 arXive:0712.139, 2008. CIRCLE PACKING 50
"... Abstract. We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinitevolume hyperbolic manifolds, with error terms which are uniform as the lattice moves through “congruence ” subgroups. We give the following application to ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We develop novel techniques using only abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinitevolume hyperbolic manifolds, with error terms which are uniform as the lattice moves through “congruence ” subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c, d) = c2 + d2, but restrict (c, d) to the orbit O = (0, 1)Γ, where Γ is an infiniteindex nonelementary finitelygenerated subgroup of SL(2, Z) containing unipotent elements. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R> 4/(δ−θ), where θ> 1/2 is the spectral gap and δ < 1 is the Hausdorff dimension of the limit set of Γ. If δ> 149/150, then we can take θ = 5/6, giving R = 25. The limit of this method is R = 9 for δ − θ> 4/9. This is the same number of prime factors as attained in Brun’s original attack on the twin prime conjecture. 1.
Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups
, 1997
"... This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topolog ..."
Abstract

Cited by 15 (0 self)
 Add to MetaCart
(Show Context)
This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H 3 /Γ is geometrically finite. We show if Γn → Γ strongly, then: (a) Mn = H 3 /Γn is geometrically finite for all n ≫ 0, (b) Λn → Λ in the Hausdorff topology, and (c) H. dim(Λn) → H. dim(Λ), if H. dim(Λ) ≥ 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H. dim(Λ) < 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higherdimensional manifolds. Applications are
Strong spectral gaps for compact quotients of products of PSL(2
 R). J. Eur. Math. Soc
"... Abstract. The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Abstract. The existence of a strong spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G has no compact factors then for general lattices a strong spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the strong spectral gap for an irreducible cocompact lattice Γ in G = PSL(2, R) d for d ≥ 2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting. introduction This note is concerned with the strong spectral gap property for an irreducible cocompact lattice Γ in G = PSL(2, R) d, d ≥ 2. Before stating our main result we review in some detail what is known about such spectral gaps more generally. Let G be a noncompact connected semisimple Lie group with finite center and let Γ be a lattice in G. For π an irreducible unitary representation of G on a Hilbert space H, we let p(π) be the infimum of all p such that there is a dense set of vectors v ∈ H with 〈π(g)v, v 〉 in L p (G). Thus if π is finite dimensional p(π) = ∞, while π is tempered if and only if p(π) = 2. In general p(π) can be computed from the Langlands parameters of π and for many purposes it is a suitable measure of the nontemperedness of π (if p(π)> 2). The regular representation, f(x) ↦ → f(xg), of G on L 2 (Γ\G) is unitary and if Γ\G is compact it decomposes into a discrete direct sum of irreducibles while if Γ\G is noncompact the decomposition involves also continuous integrals via Eisenstein series. In any case, let
DIOPHANTINE APPROXIMATION AND THE GEOMETRY OF LIMIT SETS IN GROMOV HYPERBOLIC METRIC SPACES
"... Abstract. In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater gene ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Diophantine approximation and the geometry of the limit set within which it takes place. Two results are also contained here which are purely geometric: a generalization of a theorem of C. J. Bishop and P. W. Jones to Gromov hyperbolic metric spaces, and a proof that the uniformly radial limit set of a group acting on a proper geodesic Gromov hyperbolic metric space has zero PattersonSullivan measure unless the group is quasiconvexcocompact. The latter is an application of a Diophantine theorem. Contents
SPECTRAL GAP FOR PRODUCTS OF PSL(2, R)
"... Abstract. The existence of a spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. The existence of a spectral gap for quotients Γ\G of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the RamanujanSelberg Conjectures. If G has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible cocompact lattice Γ in G = PSL(2, R) d for d ≥ 2 which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting. introduction This note is concerned with the strong spectral gap property for an irreducible cocompact lattice Γ in G = PSL(2, R) d, d ≥ 2. Before stating our main result we review in some detail what is known about such spectral gaps more generally. Let G be a noncompact connected semisimple Lie group with finite center and let Γ be a lattice in G. For π an irreducible unitary representation of G on a Hilbert space H, we let p(π) be the infimum of all p such that there is a dense set of vectors v ∈ H with 〈π(g)v, v 〉 in Lp (G). Thus if π is finite dimensional p(π) =∞, while π is tempered if and only if p(π) = 2. In general p(π) can be computed from the Langlands parameters of π and for many purposes it is a suitable measure of the nontemperedness of π (if p(π)> 2). The regular representation, f(x) ↦ → f(xg), of G on L2 (Γ\G) is unitary and if Γ\G is compact it decomposes into a discrete direct sum of irreducibles while if Γ\G is noncompact the decomposition involves also continuous integrals via Eisenstein series. In any case, let E denote the exceptional exponent set defined by