THE (POULSEN) SIMPLEX OF INVARIANT MEASURES

Abstract. Two new concepts, closeability with respect to a set of pe-riodic points and linkability of a set of periodic points of a dynamical system are introduced. Examples are provided to show that closeabil-ity and linkability are independent properties. Both properties together imply that the set of invariant measures is either a single periodic orbit or the Poulsen simplex — the unique non-trivial Choquet simplex in which extreme points are dense. Moreover, under these conditions ev-ery invariant measure has a generic point and an extension of Sigmund’s theorem about generic properties of invariant measures still holds. The periodic specification property implies closeability and linkability for the set of periodic points. The methods apply beyond systems with specification, because all beta-shifts, all S-gap shifts, and many other dynamical systems are closeable with respect to some linkable sets of periodic points. We study simplices of invariant measures of dynamical systems. A dy