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Geometrical Markov coding of geodesics on surfaces of constant negative curvature. (English) Zbl 0593.58033

This paper describes a natural geometrical representation of the geodesic flow on a surface M of constant negative curvature as a special flow over a shift on a space of sequences of generators of \(\pi_ 1(M)\). The base space is a sofic system (quotient of a subshift of finite type by a map which is here almost one-one), whose admissible finite blocks run through a set of shortest representatives of elements of \(\pi_ 1(M)\), each element occurring exactly once. The height function is the time for a particular lift of the geodesic to cross a certain fundamental region for \(\pi_ 1(M).\)
This representation is obtained by comparing symbolic representations of geodesic by doubly infinite strings of generators obtained in two different ways. Fix a set of lines on M corresponding to a decomposition which lifts to a fundamental region for \(\pi_ 1(M)\) acting in the universal cover, the hyperbolic disc D. Each line is labelled by a generator of \(\pi_ 1(M)\). A geodesic is labelled by the string of generators corresponding to the order in which it cuts these lines. Alternatively, lift a geodesic to D and record the ”boundary expansions” of the two endpoints. These are semi-infinite sequences of generators which label points on \(\partial D\), developed by R. Bowen and the author [Publ. Math., Inst. Hautes Etud. Sci. 50, 153-170 (1979; Zbl 0439.30033)] to generalize continued fraction expansions.
Except for surfaces with boundary, these two codings do not in general coincide but are shown to relate in a precise way respecting the dynamics.
These ideas develop work of Koebe, Morse, Artin and Nielsen from the early part of the century. R. L. Adler and L. Flatto [Contemp. Math. 26, 9-24 (1984; Zbl 0552.58026)] have obtained a similar coding in the case \(\pi_ 1(M)=SL(2,{\mathbb{Z}})\).

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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References:

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