Andrew Haas

Diophantine approximation on hyperbolic Riemann surfaces

Following the leads of H. Cohn and A. Schmidt we shall investigate geometric structures on hyperbolic Riemann surfaces for which Markoff-like theorems hold. The idea, which is similar to the approach taken by D. Sullivan in [21] is to look at the affinity of a geodesic for the noncompact end of a surface. More precisely, we consider the maximal depth a geodesic travels into a noncompact end. The spectrum of depths has the same structure as Markoffs spectrum with the correspondence given in terms of the geodesics length and topology. The upper discrete part of the spectrum is occupied by the simple closed geodesics and the lower limit value of the discrete spectrum is occupied by the geodesics that are limits of simple closed ones. The reason for this interaction between the geometry and the number theory becomes apparent when we transfer our attention to a Fuchsian group representing the hyperbolic surface. For example, consider the classical Modular group M6bz. The orbit of infinity under the action of M6bz is exactly the set of rational numbers. It is also the set of limit points which are fixed by parabolic transformations in the group. Each parabolic fixed point corresponds, in a sense, to the noncompact end on the quotient surface. As we shall see, the degree to which a number x is approximated by a rational number is directly related to the depth a geodesic with the endpoint x travels

The geometry of the hyperelliptic involution in genus two

The hyperelliptic involution of a genus two Riemann surface leaves invariant every simple closed geodesic on the surface. This property is not possessed by any other automorphism of a compact Riemann surface and is related to the effectiveness of the action of the mapping class group of a surface on Teichmuller space. A hyperelliptic involution of a genus p Riemann surface is an order two conformai automorphism of the surface that fixes 2p + 2 points. When such an automorphism of a surface exists it is known to be unique and the surface is said to be hyperelliptic. Every surface of genus two is hyperelliptic. Geodesies are defined in the complete Riemannian metric of constant cur- vature -1 (hyperbolic metric) which is uniquely determined by the conformai equivalence class of metrics for the given Riemann surface. A geodesic is simple if it has no self-intersections. We are happy to thank Bernie Maskit for reading the original draft and sug- gesting several significant improvements. To begin we will prove the following Theorem 1. Let J be the hyperelliptic involution of a genus two Riemann sur- face M. Then every simple closed geodesic on M is mapped onto itself by J. Furthermore, suppose a is a simple closed geodesic on M. If a is a dividing curve then J preserves the orientation of a and if a does not divide M then J reverses the orientation of a.

The connectivity of multicurves determined by integral weight train tracks

An integral weighted train track on a surface determines the isotopy class of an embedded closed 1-manifold. We are interested in the connectivity of the resulting 1-manifold. In general there is an algorithm for determining connectivity, and in the simplest case of a 2-parameter train track on a surface of genus one there is an explicit formula. We derive a formula for the connectivity of the closed 1-manifold determined by a 4-parameter train track on a surface of enus two which is computable in polynomial time. We also give necessary and sufficient conditions on the parameters for the resulting 1-manifold to be connected

Geodesic excursions into an embedded disc on a hyperbolic Riemann surface

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to an embedded disc on the surface, as well as the average amount of time it spends in the disc during each visit. This includes the case where the center of the disc is a cone point.

Invariant measures for certain linear fractional transformations mod 1

Explicit invariant measures are derived for a family of finite-toone, ergodic transformations of the unit interval having indifferent periodic orbits. Examples of interesting, non-trivial maps of [0, 1] for which one can readily compute an invariant measure absolutely continuous to Lebesgue measure are not easy to come by. The familiar examples are the Gauss map, the backward continued fraction map, and other very special cases which are close in form to the first two. See [1], [3], and [4] for an overview of the literature. Maps for which the invariant measure is infinite are even less in evidence. We will consider a family of mappings Tk,n of the unit interval that are essentially finite-to-one analogues of the backward continued fraction maps Tk = 〈 1 uk(1−x) 〉 studied in [4], where uk = 4 cos π k+2 and 〈x〉 is the fractional part of x. Both Tk and Tk,n are Mobius transformations mod 1 having indifferent periodic orbits of period k containing zero. Surprisingly, for fixed k, Tk,n converges uniformly to Tk on compact subsets of [0, 1). An explicit formula will be given for a Tk,n-invariant measure that is absolutely continuous to Lebesgue measure on [0, 1]. The measure is infinite and the density ρk,n is C∞ in the complement of the indifferent periodic orbit. Also, ρk,n converges to the Tk-invariant density ρk derived in [4]. Using Thaler’s analysis in [6] of mappings of [0, 1] with indifferent fixed points, it is possible to show that the maps Tk,n are ergodic with respect to Lebesgue measure. From now on suppose that k > 0 and n > 1 in N have been fixed and that if k = 1, then n > 2. We begin by defining the Mobius transformation that will determine Tk,n. Write Aα(x) = −αx− nα + α (n− α− 1)x− n + 1 . Then Aα(0) = α, Aα(1) = n, and A−1 α (x) = (n−1)x−nα+α (n−α−1)x+α . Aα has the following properties. Received by the editors January 1, 1998. 1991 Mathematics Subject Classification. Primary 11J70, 58F11, 58F03.

Excursion and return times of a geodesic to a subset of a hyperbolic Riemann surface

We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to a subsurface with geodesic boundary. As a consequence we get the average time a generic geodesic spends in such a subsurface. Related results are obtained for excursions into a collar neighborhood of a simple closed geodesic and the associated distribution of excursion depths.