Cross Section Maps for Geodesic Flows, I
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In this paper we investigate the relationship between two topics which at first sight seem unrelated. The first deals with ergodic properties of geodesic flows on two-dimensional surfaces of constant negative curvature, a rather active area in the thirties studied by many well known mathematicians. For a detailed survey of the work during that period see [H2]. The second one deals with ergodic properties of noninvertible mappings of the unit interval, a current popular subject and one also with an interesting history going back to Gauss (See [B]). We shall show how each of these subjects sheds light on the other. Ergodic properties of interval maps can be used to prove ergodicity of the flows and conversely. Furthermore, explicit formulas for invariant measures of interval maps can be derived from the invariance of hyperbolic measure for the flows. (Actually we could go a step further and trace a connection of these formulas to Liouville’s theorem for Hamiltonian Systems.) This fact is particularly interesting as there is a paucity of explicit formulas for invariant measures of interval maps and we have here a method of deriving a class of these. In particular we shall show how Gauss’s formula for the invariant measure associated with continued fractions, which seems to have been produced ad hoc, can be derived anew.
KeywordsInvariant Measure Fuchsian Group Geodesic Flow Ergodic Property Fractional Linear Transformation
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