The connection matrix in Morse-Smale flows

Abstract

In a Morse-Smale flow with no periodic orbits, it is shown that the connection matrix is unique. In the case of periodic orbits, nonuniqueness can occur. We show that on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifolds, with some technical assumptions, given a connection matrix for the flow, one can replace the periodic orbits with doubly-connected rest points and obtain a new flow with no periodic orbits having the given connection matrix.