Bounded geodesics of Riemann surfaces and hyperbolic manifolds

Abstract

We study the set of bounded geodesies of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.

Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces

“…Definition 2. 35. Let X be a hyperbolic metric space, and let (x n ) ∞ 1 be a sequence in X converging to a point ξ ∈ ∂X.…”

“…Beginning with Patterson, many authors [6,27,50,69,79,80,82,90,91,92] have considered generalizations of these theorems to the case of a nonelementary geometrically finite group acting on standard hyperbolic space X = H d+1 for some d ∈ N, considering a parabolic fixed point if G has at least one cusp, and a hyperbolic fixed point otherwise. Moreover, many theorems in the literature concerning the asymptotic behavior of geodesics in geometrically finite manifolds [2,3,23,35,63,64,81,84] can be recast in terms of Diophantine approximation. To date, the most general setup is that of S. D. Hersonsky and F. Paulin, who generalized Dirichlet and Khinchin's theorems to the case of pinched Hadamard manifolds [46,47] and proved an analogue of Khinchin's theorem for uniform trees [48]; nevertheless, they still assume that the group is geometrically finite, and they only approximate by parabolic orbits.…”

On Badly Approximable Complex Numbers

“…Hence, we suppose that n > 0 and proceed by induction. We have already avoided counter-examples to equation (6) earlier in the game, and hence we need only worry about q ∈ O D for which Hence, the maximal number k of disjoint B i that can be contained in (1 + 2αr n )C is bounded above by…”

“…Among other things, he proved that Bad 1 = ∅, and obtained quantitative information on the set of best possible values for K(z) as z varies. When the set of complex numbers is considered as the limit set of the Picard group, it can be derived from results of Bishop and Jones [3] or Fernandez and Melián [6] that the set Bad 1 has maximal Hausdorff dimension. This is also proved in [4] by more elementary means, using the framework of Schmidt games (see below).…”

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

“…It is well-known that isoperimetric inequalities are of interest in applied and pure mathematics [22,48]. The Cheeger isoperimetric inequality is related with many conformal invariants in Riemannian manifolds and graphs, namely the exponent of convergence, the bottom of the spectrum of the Laplace-Beltrami operator, Poincaré-Sobolev inequalities, and the Hausdorff dimensions of the sets of both escaping and bounded geodesics in negatively curved surfaces [4,13], [18, p. 228], [23,[27][28][29][30][31]43,46], [52, p. 333]. Isoperimetric inequality is also closely related to Ancona's project on the space of positive harmonic functions of Gromov-hyperbolic graphs and manifolds [7][8][9].…”