2,582 research outputs found
Perfect Reeb flows and action-index relations
We study non-degenerate Reeb flows arising from perfect contact forms, i.e.,
the forms with vanishing contact homology differential. In particular, we
obtain upper bounds on the number of simple closed Reeb orbits for such forms
on a variety of contact manifolds and certain action-index resonance relations
for the standard contact sphere. Using these results, we reprove a theorem due
to Bourgeois, Cieliebak and Ekholm characterizing perfect Reeb flows on the
standard contact three-sphere as non-degenerate Reeb flows with exactly two
simple closed orbits.Comment: 15 page
Periodic Orbits of Hamiltonian Systems Linear and Hyperbolic at Infinity
We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by
compactly supported time-dependent perturbations of hyperbolic quadratic forms.
We prove that, under some natural assumptions, such a diffeomorphism must have
simple periodic orbits of arbitrarily large period when it has fixed points
which are not necessary from a homological perspective.Comment: 21 pages; substantially revised, final version; to appear in Pacific
Journal of Mathematic
Nonconcentration of return times
We show that the distribution of the first return time to the origin,
v, of a simple random walk on an infinite recurrent graph is heavy tailed and
nonconcentrated. More precisely, if is the degree of v, then for any
we have and
for some
universal constants and . The first bound is attained for all t
when the underlying graph is , and as for the second bound, we
construct an example of a recurrent graph G for which it is attained for
infinitely many t's. Furthermore, we show that in the comb product of that
graph G with , two independent random walks collide infinitely many
times almost surely. This answers negatively a question of Krishnapur and Peres
[Electron. Commun. Probab. 9 (2004) 72-81] who asked whether every comb product
of two infinite recurrent graphs has the finite collision property.Comment: Published in at http://dx.doi.org/10.1214/12-AOP785 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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