arXiv subjects
Publications and source records attributed to Christian Pries.
We give a short discussion about a weaker form of minimality (called quasi-minimality). We call a system quasi-minimal if all dense orbits form an open set. It is hard to find examples which are not already minimal. Since elliptic behaviour makes them minimal, these systems are regarded as parabolic systems. Indeed, we show that a quasi-minimal homeomorphism on a manifold is not expansive (hyperbolic).
The theorem of Le Calvez and Yoccoz states that there are no minimal homeomorphisms on the finite punctered 2-dimensional sphere S 2 . We show that this does not hold for other surfaces. Moreover, we discuss why the fast-conjugation-method fails in the most cases to construct such homeomorphisms. This article based on an old unpublished article (Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits) with incorrect results.
The author shows that equicontinuous geodesic flows on surfaces are periodic. A similar result for flows on 3-manifolds is also proven. The idea of the proof is to show that the return map is recurrent and therefore periodic.
The paper shows that the curvature of RP2 is constant iff all geodesics are closed. Therefore RP2 is the first known manifold with only one G-structure. It took quiete a long time to find such a manifold. The author shows only that if all geodesics are closed then there are infinitely many simple closed geodesics. This proof is based on the geodesic return map and the theory of topological dynamics. From important results of Green, Grove and Gromoll one can conclude the theorem.