 
Summary: MINIMAL FLOWS ON MULTIPUNCTURED SURFACES OF
INFINITE TYPE
KONSTANTIN ATHANASSOPOULOS AND ANTONIOS MANOUSSOS
A multipunctured surface is an open 2manifold obtained from a closed 2
manifold M by removing a nonempty, closed and totally disconnected set F. The
multipunctured surface M\F will be called of finite type if F is finite, and of infinite
type otherwise. In this short note we study the behaviour of the orbits of a given
minimal flow on a multipunctured surface M\F at infinity, that is, near the points of
F. Any flow on M\F has an extension to a flow on M that fixes F pointwise [1, Satz
2.3]. In view of the work of C. Gutierrez [5], there is no loss of generality if we assume
that everything is smooth.
Examples of minimal flows on multipunctured surfaces of finite type are given in
[4] and [7], and from these one can obtain minimal flows on multipunctured surfaces
of infinite type, by multiplying the infinitesimal generator with a suitable smooth
function. The aim of the present note is to show that every minimal flow on a
multipunctured surface of infinite type is obtained in this way; see Theorem 4 below.
In thefinitetype case, every point of Fhas to be a possibly degenerate saddle, and
it follows from the PoincareHopf Index Theorem that the number of orbits in M\F
with empty positive (negative) limit set in M\.Fis equal to \F\ x(M), where /(M) is
the Euler characteristic of M. So if F is a finite subset of the torus T2
