Symbolic dynamics for hyperbolic surfaces of finite area.

摘要

The objective of this work is the study of the geodesic flow for noncompact hyperbolic surfaces of finite area. The geodesic flow for compact surfaces has been much studied. For finite area noncompact surfaces, however, most of the interesting dynamics still remains.; It is well known that a hyperbolic surface arises m the quotient space $G\backslash H2$, where $H2$ is the hyperbolic plane and $G$ is a group of isometries of the plane into itself. We define, then, a cusped surface as a surface of finite area for which $G$ has a finite number of primitive parabolic elements up to conjugacy. Each of these elements corresponds to one cusp on the hyperbolic surface. The pictorial model of a cusp is a half-cylinder $S1\times 1,\infty$ with the length of the horocircle $S1\times t$ decreasing exponentially with t. Therefore cusps make the surface unbounded $G\backslash H2$ but add only a finite area.; In this scenario we define the geodesic flow on the unit tangent bundle of $G\backslash H2,yt:G\backslash SH2\to G\backslash \; SH2$. The natural way of studying this flow is through its lift to a geodesic flow whose flowlines consist of unit tangent vectors to geodesics on $H2$. The techniques used to understand these systems are based on Markov partitions and subshifts of finite type. Our aim is to introduce them using geometrical methods. We restrict our attention to surfaces with only one cusp, although we expect that the situation with a finite number of cusps can be treated in a similar fashion.; We consider a fundamental region on $H2$ for $G$ as described by Ford. This domain allows us to construct a canonical ideal tessellation of $H2$ by hyperbolic polygons (by ideal we mean that all the vertices of these polygons lie on the circle at infinity). With this tessellation in hand we define a Markov system of interval maps which codes the geodesic flow. Not only that but we also get a finite area Markov section on $G\backslash SH2$, known as a necktie. It can be visualized as a cylindrical region with a zig-zag shaped bottom.; The Markov section gives rise to a coded description of the geodesic flow known as symbolic dynamics. As an intuitive description of this code, suppose a symbol is given for each side of each polygon in the tessellation of $H2$. Each geodesic is associated to the sequence of symbols of the sides that it crosses. A geodesic can be reconstructed from this sequence of symbols. Likewise the symbol sequences which arise from geodesics admit an explicit description.