In 1971, M. M. Peixoto [15] introduced an important
topological
invariant of Morse–Smale flows on surfaces, which he called a
distinguished
graph $X^*$ associated with a given flow.
Here we show how the Peixoto invariant 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
$R$ is 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 flowsAnd foliations, see
[3]. We do not treat
the case of foliations, bearing in mind that they are defined by involutive
flows on covering manifolds [9]..
As a counterpart to the equivalence 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 idea of ‘rotation systems’ on graphs can be further exploited
in the
study of recurrent flows (and foliations) with several disjoint quasiminimal
sets on surfaces [10].