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We study invariant measures of a dynamical system given by a
continuous map on a compact metric space. Ergodic measures are
extreme points of the simplex of all invariant measures and every
invariant measure is a generalized affine combination of ergodic
ones. Hence, the set of invariant measures is a Choquet simplex. The
Poulsen simplex is an universal object among all Choquet
simplices. It is characterized as the unique infinite-dimensional
Choquet simplex in which extreme points are dense. This brings us to
the question: When the set of invariant measures of a
dynamical system is the Poulsen simplex? This problem has been
considered by many authors. Sigmund proved (among many others
things) that it is enough to assume that system has the periodic
specification property. Examples of such systems include mixing
SFT's and sofic shifts, mixing axiom A diffeomorphisms, mixing
continuous interval (graph) maps, and geodesic flows on manifolds
with negative curvature. Many authors have weakened the
specification property to apply similar techniques to a wider range
of examples. An (incomplete) list contains such names as Climenhaga,
Coudène, Dateyama, Gelfert, Hofbauer, Oliveira, Pfister,
Varandas and Yamamoto (and their co-authors). Our work is very much
in the spirit of these developments. We define yet another
specification-like property and construct examples to examine
limitations of various approaches. The main result describes
properties of the simplex of invariant measures under our
specification-like assumption.
It turns out that the notion of Besicovitch pseudometric is a very
useful tool in these investigations. As a main technical tool we
introduce the theory of pseudorbits with respect to the Besicovitch
pseudometric and measures generated by them. Our version extends the
Sigmund theorem and can be applied to examples lying beyond the
scope of previously known results.
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