15,055 research outputs found
Invariant measures for Cherry flows
We investigate the invariant probability measures for Cherry flows, i.e.
flows on the two-torus which have a saddle, a source, and no other fixed
points, closed orbits or homoclinic orbits. In the case when the saddle is
dissipative or conservative we show that the only invariant probability
measures are the Dirac measures at the two fixed points, and the Dirac measure
at the saddle is the physical measure. In the other case we prove that there
exists also an invariant probability measure supported on the quasi-minimal
set, we discuss some situations when this other invariant measure is the
physical measure, and conjecture that this is always the case. The main
techniques used are the study of the integrability of the return time with
respect to the invariant measure of the return map to a closed transversal to
the flow, and the study of the close returns near the saddle.Comment: 12 pages; updated versio
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
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