Minimal Flows on Multipunctured Surfaces of Infinite Type

Abstract

A multipunctured surface is an open 2-manifold obtained from a closed 2manifold 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 the finite type case, every point of F has 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 T, then every minimal flow on T\F is constructed (modulo topological equivalence) from an irrational vector field by multiplication with a smooth function which vanishes exactly on F. Using the examples in [4], it is not hard to see that if x(M) < 0, then all possible cases of behaviour at infinity of minimal flows on multipunctured surfaces of finite type M\F can occur. We turn now to the study of minimal flows on multipunctured surfaces of infinite type. We shall use the machinery of isolated invariant sets and isolating blocks developed by C. Conley and R. Easton in [3]. Recall that a compact invariant set A of a given smooth flow on a manifold M is called isolated if it is the maximal invariant set in some of its compact neighbourhoods. An isolating block N is a compact submanifold with boundary of M, of the same dimension, such that the boundary dN of N is the union of two submanifolds n, n~ of codimension 1 in M, with common boundary x such that the flow is transverse into N on «\T, transverse out of N on «\T and externally tangent to N on T. It is proved in [3] that every neighbourhood of an isolated invariant set A contains an isolating block N having A in its interior, and A is the maximal invariant set in N. If M\F is a multipunctured surface carrying a minimal flow, then F is an isolated invariant set with respect to the extended flow on M. Moreover, every point of F has arbitrarily small compact neighbourhoods in F which are isolated invariant sets, because F is totally disconnected. By the classification of the noncompact surfaces