Remote limit points on surfaces (Q1867252)

The authors study the omega limit sets of orbits of a flow on a compact surface \(M\) of genus at least 2. The flow on \(M\) lifts to the universal covering \(\widetilde{M}\) of \(M\). The orbits studied are those with lifts to \(\widetilde{M}\) that limit to a fixed point \(a\in \mathbb{K}\) of a covering transformation \(T\). The circle orthogonal to \(\mathbb{K}\) determined by the fixed points \(a\) and \(b\) of \(T\) is called the axis of \(T\). The remote limit points are the points in the omega limit set arising from the sequences of points on the positive orbit for which the hyperbolic distance of the lift to the axis of \(T\) goes to infinity. The main result about these remote limit points is that they must be fixed points.