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In this work we will consider a class of second order perturbed Hamiltonian systems of the form , where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained...
Let Gbar = G{nt, nt | nt+1, t ≥ 0} be a subgroup of all roots of unity generated by exp(2πi/nt}, t ≥ 0, and let τ: (X, β, μ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle φ, φ: X → Z2, admitting an approximation with speed 0(1/n1+ε, ε>0) there exists a Morse cocycle φ such that the corresponding transformations τφ and τψ are relatively isomorphic. An effective way of a construction of the Morse cocycle φ is given. There is a cocycle φ oddly approximated with...
We investigate the natural domain of definition of the Godbillon-Vey 2- dimensional cohomology class of the group of diffeomorphisms of the circle. We introduce the notion of area functionals on a space of functions on the circle, we give a sufficiently large space of functions with nontrivial area functional and we give a sufficiently large group of Lipschitz homeomorphisms of the circle where the Godbillon-Vey class is defined.
A certain family of homogeneous spaces is investigated. Basic invariant operators for each of these structures are presented and some analogies to Levi-Civita connections of Riemannian geometry are pointed out.
We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus with a complex line , with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to . The converse statement is also true, namely a holomorphic bundle on which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....
For Schrödinger operator on Riemannian manifolds with conical end, we study the contribution of zero energy resonant states to the singularity of the resolvent of near zero. Long-time expansion of the Schrödinger group is obtained under a non-trapping condition at high energies.
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