291 research outputs found
Successive Minima and Best Simultaneous Diophantine Approximations
We study the problem of best approximations of a vector by rational vectors of a lattice whose
common denominator is bounded. To this end we introduce successive minima for a
periodic lattice structure and extend some classical results from geometry of
numbers to this structure. This leads to bounds for the best approximation
problem which generalize and improve former results.Comment: 8 page
Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
Invariant tori play a fundamental role in the dynamics of symplectic and
volume-preserving maps. Codimension-one tori are particularly important as they
form barriers to transport. Such tori foliate the phase space of integrable,
volume-preserving maps with one action and angles. For the area-preserving
case, Greene's residue criterion is often used to predict the destruction of
tori from the properties of nearby periodic orbits. Even though KAM theory
applies to the three-dimensional case, the robustness of tori in such systems
is still poorly understood. We study a three-dimensional, reversible,
volume-preserving analogue of Chirikov's standard map with one action and two
angles. We investigate the preservation and destruction of tori under
perturbation by computing the "residue" of nearby periodic orbits. We find tori
with Diophantine rotation vectors in the "spiral mean" cubic algebraic field.
The residue is used to generate the critical function of the map and find a
candidate for the most robust torus.Comment: laTeX, 40 pages, 26 figure
Support of Laurent series algebraic over the field of formal power series
This work is devoted to the study of the support of a Laurent series in
several variables which is algebraic over the ring of power series over a
characteristic zero field. Our first result is the existence of a kind of
maximal dual cone of the support of such a Laurent series. As an application of
this result we provide a gap theorem for Laurent series which are algebraic
over the field of formal power series. We also relate these results to
diophantine properties of the fields of Laurent series.Comment: 31 pages. To appear in Proc. London Math. So
Rational approximations, multidimensional continued fractions and lattice reduction
We first survey the current state of the art concerning the dynamical
properties of multidimensional continued fraction algorithms defined
dynamically as piecewise fractional maps and compare them with algorithms based
on lattice reduction. We discuss their convergence properties and the quality
of the rational approximation, and stress the interest for these algorithms to
be obtained by iterating dynamical systems. We then focus on an algorithm based
on the classical Jacobi--Perron algorithm involving the nearest integer part.
We describe its Markov properties and we suggest a possible procedure for
proving the existence of a finite ergodic invariant measure absolutely
continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure
Renormalisation scheme for vector fields on T2 with a diophantine frequency
We construct a rigorous renormalisation scheme for analytic vector fields on
the 2-torus of Poincare type. We show that iterating this procedure there is
convergence to a limit set with a ``Gauss map'' dynamics on it, related to the
continued fraction expansion of the slope of the frequencies. This is valid for
diophantine frequency vectors.Comment: final versio
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