GRAPHS AND FLOWS ON SURFACES

In 1971, M. M. Peixoto [15] introduced an important topological invari ant of Morse-Smale flows on surfaces, which he called a distinguished graph X associated with a given flow. Here we show how the Peixoto in variant can be essentially simplified and revised by adopting a purely topological point of view connected with the embeddings of arbitrary graphs into compact surfaces. The newly obtained invariant, X-R, is a rotation of a graph X generated by a Morse-Smale flow. (a rotation R i s a cyclic order of edges given in every vertex of X.) The invariant X -R 'reads-off' the topological information carried by a flow, being in a one-to-one correspondence with the topological equivalence classes of Morse-Smale flows double dagger. As a counterpart to the equivalenc e result we prove a realization theorem for an 'abstractly given' X-R. (Our methods are completely different from those of Peixoto and they clarify the connections between graphs and flows on surfaces.) The ide a of 'rotation systems' on graphs can be further exploited in the stud y of recurrent flows (and foliations) with several disjoint quasiminim al sets on surfaces [10].