Periodic orbits and P stable orbit closures of continuous flows on surfaces (Q1122199)

This paper deals with the problem of the coexistence of non-zero- homotopic periodic orbits and Poisson stable orbits for flows on surfaces. It is known that on the torus they are incompatible. Let f be a continuous flow on the closed surface M of genus g, \(g\geq 2\), \(r_ 0\) the number of distinct Poisson stable orbit closures, r the number of non-zero-homotopic to each other (in the nonorientable case let \(r_ 1\) \((r_ 2)\) denote the number of such orbits, which, in addition, are one (two)-sided). The main result consists in the following two inequalities: \(r_ 0+r\leq 3g-3,\) if M is orientable, and \(4r_ 0+r_ 1+r_ 2\leq 3g-3\) in the nonorientable case. In both cases equalities are achieved. [See also \textit{N. G. Markley}, Proc. Am. Math. Soc. 25, 413-416 (1970; Zbl 0198.569).]