Recurrent orbits for flows on surfaces

A flow on a surface M lifts to a flow on , the universal cover of M. For a compact surface of genus at least 2, is the interior of the unit disk and the covering transformations are a group of hyperbolic linear fractional transformations with two fixed points on the unit circle, . The analysis in this paper will be based almost entirely on geometric properties of the lifted flow and homotopy properties of loops constructed from local cross sections and orbits. The orbits studied have at least one nonfixed point in their ω-limit set and have lifts to that limit to a point of . The main results describe the structure of the ω-limit sets of such orbits. In Theorem 8 it is shown that all nonfixed ω-limit points are “isolated” if and only if the lifted orbit limits to a fixed point of a covering transformation, which is called a rational point. In Theorem 13 sufficient conditions on the ω-limit set are established for the existence of positively and negatively recurrent points in the omega limit set of an orbit. This result is then used in Section 6 to give new geometric proofs to two of A. Maierʹs classic results about the existence of positively recurrent orbits in omega limit sets.