6 research outputs found
The Conley Conjecture and Beyond
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur
Random Chain Complexes
We study random, finite-dimensional, ungraded chain complexes over a finite
field and show that for a uniformly distributed differential a complex has the
smallest possible homology with the highest probability: either zero or
one-dimensional homology depending on the parity of the dimension of the
complex. We prove that as the order of the field goes to infinity the
probability distribution concentrates in the smallest possible dimension of the
homology. On the other hand, the limit probability distribution, as the
dimension of the complex goes to infinity, is a super-exponentially decreasing,
but strictly positive, function of the dimension of the homology