Related papers: An elegant model of the geodesic flow on the modul…
We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions,…
In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in…
In this article, we characterize two kinds of exceptional orbits of the geodesic flow associated with the Modular surface in terms of a two-parameter family of continued fraction expansion of endpoints of the lifts to the hyperbolic plane…
The connection between cutting sequences of geodesics on the modular surface $\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$ and regular continued fractions was established by Series, and Heersink expanded the cross-section of the…
In this note we study the thermodynamic formalism for the positive geodesic flow on the modular surface. We define the pressure and prove the variational principle. We also establish conditions for the the pressure to be real analytic and…
We extend the Series' connection between the modular surface $\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions…
The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $\Gamma\backslash…
In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular…
This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2,Z) with respect to the standard fundamental domain F = {z=x+iy: -1/2 <= x <= 1/2 and |z|>=1} of PSL(2,Z). The cutting sequence for a vertical…
In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…
The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.
Gauss' classical reduction theory for indefinite binary quadratic forms over $\mathbb{Z}$ has originally been proven by means of purely algebraic and arithmetic considerations. It was later discovered that this reduction theory is closely…
Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to…
In this note we give a criterion for the existence of a fractional-linear integral for a geodesic flow on a Riemannian surface and explain that modulo M\"obius transformations the moduli space of such local integrals (if nonempty) is either…
We describe a general method of arithmetic coding of geodesics on the modular surface based on a two parameter family of continued fraction transformations studied previously by the authors. The finite rectangular structure of the…
We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…
We establish the background for the study of geodesics on noncompact polygonal surfaces. For illustration, we study the recurrence of geodesics on $Z$-periodic polygonal surfaces. We prove, in particular, that almost all geodesics on a…
It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are very abstract and not easy to understand. To establish a more…
Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the…
We prove several results concerning the existence of surfaces of section for the geodesic flows of closed orientable Riemannian surfaces. The surfaces of section $\Sigma$ that we construct are either Birkhoff sections, meaning that they…