293 research outputs found
Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type
We describe the topological behavior of typical orbits of complex quadratic
polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type.
Here we prove that for such Brjuno values of alpha the closure of the critical
orbit, which is the measure theoretic attractor of the map, has zero area. Then
combining with Part I of this work, we show that the limit set of the orbit of
a typical point in the Julia set is equal to the closure of the critical orbit.Comment: 38 pages, 5 figures; fixed the issues with processing the figure
Linear stability analysis of resonant periodic motions in the restricted three-body problem
The equations of the restricted three-body problem describe the motion of a
massless particle under the influence of two primaries of masses and
, , that circle each other with period equal to
. When , the problem admits orbits for the massless particle that
are ellipses of eccentricity with the primary of mass 1 located at one of
the focii. If the period is a rational multiple of , denoted ,
some of these orbits perturb to periodic motions for . For typical
values of and , two resonant periodic motions are obtained for . We show that the characteristic multipliers of both these motions are given
by expressions of the form in the limit . The coefficient is analytic in at and
C(e,p,q)=O(e^{\abs{p-q}}). The coefficients in front of e^{\abs{p-q}},
obtained when is expanded in powers of for the two resonant
periodic motions, sum to zero. Typically, if one of the two resonant periodic
motions is of elliptic type the other is of hyperbolic type. We give similar
results for retrograde periodic motions and discuss periodic motions that
nearly collide with the primary of mass
Resonances and O-curves in Hamiltonian systems
We investigate the problem of the existence of trajectories asymptotic to
elliptic equilibria of Hamiltonian systems in the presence of resonances.Comment: 12 page
Eigenfunction statistics for a point scatterer on a three-dimensional torus
In this paper we study eigenfunction statistics for a point scatterer (the
Laplacian perturbed by a delta-potential) on a three-dimensional flat torus.
The eigenfunctions of this operator are the eigenfunctions of the Laplacian
which vanish at the scatterer, together with a set of new eigenfunctions
(perturbed eigenfunctions). We first show that for a point scatterer on the
standard torus all of the perturbed eigenfunctions are uniformly distributed in
configuration space. Then we investigate the same problem for a point scatterer
on a flat torus with some irrationality conditions, and show uniform
distribution in configuration space for almost all of the perturbed
eigenfunctions.Comment: Revised according to referee's comments. Accepted for publication in
Annales Henri Poincar
Magnetic flows on Sol-manifolds: dynamical and symplectic aspects
We consider magnetic flows on compact quotients of the 3-dimensional solvable
geometry Sol determined by the usual left-invariant metric and the
distinguished monopole. We show that these flows have positive Liouville
entropy and therefore are never completely integrable. This should be compared
with the known fact that the underlying geodesic flow is completely integrable
in spite of having positive topological entropy. We also show that for a large
class of twisted cotangent bundles of solvable manifolds every compact set is
displaceable.Comment: Final version to appear in CMP. Two new remarks have been added as
well as some numerical calculations for metric entrop
Weyl group multiple Dirichlet series constructed from quadratic characters
We construct multiple Dirichlet series in several complex variables whose
coefficients involve quadratic residue symbols. The series are shown to have an
analytic continuation and satisfy a certain group of functional equations.
These are the first examples of an infinite collection of unstable Weyl group
multiple Dirichlet series in greater than two variables.Comment: incorporated referee's comment
Representations of integers by certain positive definite binary quadratic forms
We prove part of a conjecture of Borwein and Choi concerning an estimate on
the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.Comment: 8 pages, submitte
Higher Spin Fields in Siegel Space, Currents and Theta Functions
Dynamics of four-dimensional massless fields of all spins is formulated in
the Siegel space of complex symmetric matrices. It is shown that
the unfolded equations of free massless fields, that have a form of
multidimensional Schrodinger equations, naturally distinguish between positive-
and negative-frequency solutions of relativistic field equations, i.e.
particles and antiparticles. Multidimensional Riemann theta functions are shown
to solve massless field equations in the Siegel space. We establish the
correspondence between conserved higher-spin currents in four-dimensional
Minkowski space and those in the ten-dimensional matrix space. It is shown that
global symmetry parameters of the current in the matrix space should be
singular to reproduce a nonzero current in Minkowski space. The \D-function
integral evolution formulae for 4d massless fields in the Fock-Siegel space are
obtained. The generalization of the proposed scheme to higher dimensions and
systems of higher ranks is considered.Comment: LaTeX, 38 pages, v.3: clarifications, acknowledgements and references
added, typos corrected, v.4: more comments and references added, typos
corrected, the version to appear in JHE
On the most compact regular lattice in large dimensions: A statistical mechanical approach
In this paper I will approach the computation of the maximum density of
regular lattices in large dimensions using a statistical mechanics approach.
The starting point will be some theorems of Roger, which are virtually unknown
in the community of physicists. Using his approach one can see that there are
many similarities (and differences) with the problem of computing the entropy
of a liquid of perfect spheres. The relation between the two problems is
investigated in details. Some conjectures are presented, that need further
investigation in order to check their consistency.Comment: 27 page
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