Local connectivity of limit sets (Q1880860)

Some results on the dynamics of continuous flows on the torus from \textit{N. G. Markley} [Mich. Math. J. 25, 45--52 (1978; Zbl 0396.58028)] are extended to a compact orientable surface \(M\) of genus greater than one. It is shown that if there is a positive orbit on \(M\) such that its lift to the Poincaré disk limits to a rational point and either its \(\omega\)-limit set is locally connected or the set of fixed-points in its \(\omega\)-limit set is totally disconnected, then the \(\omega\)-limit set contains an invariant simple closed curve that is not null homotopic. Further, sufficient conditions are found for the orbit's lift to stay a bounded distance from a geodesic with the same limiting point.