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Symbolic dynamics and geodesic flows. (English) Zbl 0770.58011

Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 10, Année 1991-1992, 109-129 (1992).
This paper is a survey on the basic symbolic dynamics approach to geodesic flows on compact manifolds with negative sectional curvature and is mainly based on the following two references: W. Parry and the author [Zeta functions and the periodic orbit structure of hyperbolic dynamics (1990; Zbl 0726.58003)] and T. Bedford, M. Keane, and C. Series [Ergodic theory, symbolic dynamics and hyperbolic spaces (1989; Zbl 0743.00040)].
The basic tool in dynamical systems is the use of a family of Markov sections, which are a particular class of Poincaré sections, and thanks to them, the Poincaré map is reduced into a symbolic model where the symbols are the indices of the Markov sections and second the whole flow is also reduced into a symbolic model.
The main proofs are sketched, for example, the existence of a family of Markov sections for a geodesic flow, the Ruelle operator theorem or the theorem on the domain of the zeta function.

MSC:

37E99 Low-dimensional dynamical systems
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
54C70 Entropy in general topology
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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