In the classical Poincaré–Bendixson theory
the object of study are the limit sets of a continuous flow on the 2-sphere
S2 and the behaviour of the orbits near them
(see [7, 9]). In [2] the
second author proved that an assertion similar to the Poincaré–Bendixson
theorem is true in the wider class of the 1-dimensional invariant (internally)
chain recurrent continua of flows on S2. On the other
hand, it is known that among the closed 2-manifolds, the 2-sphere S2,
the projective plane RP2 and the
Klein bottle K2 are the only ones for which the
Poincaré–Bendixson theorem is true
(see [1, 8, 11]).
The motivation of the present paper was to examine to what extent the main
results of [2] carry over to flows on RP2
and K2. A first attempt to study chain
recurrent sets of flows on closed 2-manifolds other than the 2-sphere
was [3]. As one
expects, the results of [2] carry over easily to
RP2, since chain recurrence behaves
well with respect to regular covering maps of compact manifolds, as we show in
Section 3. The situation with K2 is quite different,
since it is doubly covered by the
2-torus T2, where we have no Poincaré–Bendixson
theorem. Actually, the Poincaré–Bendixson theorem for
1-dimensional invariant chain recurrent continua of flows on
K2 is not true. For example, identifying
suitably the boundary periodic orbits of
a 2-dimensional Reeb flow on a closed annulus (see
[7, chapter III, 2·6]) we get a
flow on K2 with a 1-dimensional invariant chain
recurrent continuum consisting of
the unique periodic orbit and another orbit, which spirals against it in positive and
negative time. As we prove in Theorem 4·4, this situation, or concatenations of it,
is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant
chain recurrent continua of flows on K2 is not
true. Then, we are concerned with
the topological structure of the 1-dimensional chain components of a flow on
K2 with finitely many singularities. In
Proposition 4·6 we find when such a set consists
of finitely many orbits and is homeomorphic to a finite graph. An example shows
that the hypothesis of Proposition 4·6 is essential. Finally,
in Theorem 4·9 we give a
description of the structure of the 1-dimensional chain components of a
flow on K2 with finitely many singular points.