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MINIMAL FLOWS ON MULTIPUNCTURED SURFACES OF INFINITE TYPE
 

Summary: MINIMAL FLOWS ON MULTIPUNCTURED SURFACES OF
INFINITE TYPE
KONSTANTIN ATHANASSOPOULOS AND ANTONIOS MANOUSSOS
A multipunctured surface is an open 2-manifold 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 Poincare-Hopf 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

  

Source: Athanassopoulos, Konstantin - Department of Mathematics, University of Crete
Manoussos, Antonios - Fakultät für Mathematik, Universität Bielefeld

 

Collections: Mathematics