Symbolic dynamics for geodesic floes

Abstract

The subject of symbolic dynamics is of central importance in the modern theory of dynamical systems. The use of symbolic sequences to study dynamical properties of geodesies originates in the work of Koebe [21, 22] and Morse [24, 25] and is already foreshadowed in Hadamard [15] and Jordan [19]. The method of Koebe and of Morse is to code a geodesic on a surface M of negative curvature by recording the order in which it traverses a given set of labelled curves on M. The treatment of Morse allows variable curvature but assumes at least two boundary components, whereas Koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface. This last case is handled by recording crossings of a fixed pants decomposition, anticipating the Thurston parameterization of simple curves as described in [11]. Both Koebe and Morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense (transitive) geodesies. Morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion (in modern terminology, a minimal nowhere dense set). Later [26] Morse treated the special closed surfaces of genus g associated to tesselations of the disc D by regular 4</-gons. Each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin-face M. The set of isometries appearing as labels generate TTI(M). Thus any geodesic is coded as a doubly infinite sequence of generators of iri(M). The same method applies quite generally to tesselations associated to any Fuchsian group. We call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric. The difficulty of course is to determine precisely the class of sequences which occur. For a surface with boundary, one obtains exactly reduced -sequences^in^the=generators^In^generaLthe=problem4s=GompliGated1=henGe=the^ difficulties encountered in [26] (see also Theorem 3 below). There is another method of coding geodesies, using certain boundary expansions for points at infinity in the universal cover of M. For the modular surface H/SL(2,2) the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the Nielsen boundary expansions of [27].