797 research outputs found
The mixmaster universe: A chaotic Farey tale
When gravitational fields are at their strongest, the evolution of spacetime
is thought to be highly erratic. Over the past decade debate has raged over
whether this evolution can be classified as chaotic. The debate has centered on
the homogeneous but anisotropic mixmaster universe. A definite resolution has
been lacking as the techniques used to study the mixmaster dynamics yield
observer dependent answers. Here we resolve the conflict by using observer
independent, fractal methods. We prove the mixmaster universe is chaotic by
exposing the fractal strange repellor that characterizes the dynamics. The
repellor is laid bare in both the 6-dimensional minisuperspace of the full
Einstein equations, and in a 2-dimensional discretisation of the dynamics. The
chaos is encoded in a special set of numbers that form the irrational Farey
tree. We quantify the chaos by calculating the strange repellor's Lyapunov
dimension, topological entropy and multifractal dimensions. As all of these
quantities are coordinate, or gauge independent, there is no longer any
ambiguity--the mixmaster universe is indeed chaotic.Comment: 45 pages, RevTeX, 19 Figures included, submitted to PR
Gaussian Behavior of Quadratic Irrationals
We study the probabilistic behaviour of the continued fraction expansion of a
quadratic irrational number, when weighted by some "additive" cost. We prove
asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal
with the underlying dynamical system associated with the Gauss map, and its
weighted periodic trajectories. We work with analytic combinatorics methods,
and mainly with bivariate Dirichlet generating functions; we use various tools,
from number theory (the Landau Theorem), from probability (the Quasi-Powers
Theorem), or from dynamical systems: our main object of study is the (weighted)
transfer operator, that we relate with the generating functions of interest.
The present paper exhibits a strong parallelism with the methods which have
been previously introduced by Baladi and Vall\'ee in the study of rational
trajectories. However, the present study is more involved and uses a deeper
functional analysis framework.Comment: 39 pages In this second version, we have added an annex that provides
a detailed study of the trace of the weighted transfer operator. We have also
corrected an error that appeared in the computation of the norm of the
operator when acting in the Banach space of analytic functions defined in the
paper. Also, we give a simpler proof for Theorem
Lubricated friction between incommensurate substrates
This paper is part of a study of the frictional dynamics of a confined solid
lubricant film - modelled as a one-dimensional chain of interacting particles
confined between two ideally incommensurate substrates, one of which is driven
relative to the other through an attached spring moving at constant velocity.
This model system is characterized by three inherent length scales; depending
on the precise choice of incommensurability among them it displays a strikingly
different tribological behavior. Contrary to two length-scale systems such as
the standard Frenkel-Kontorova (FK) model, for large chain stiffness one finds
that here the most favorable (lowest friction) sliding regime is achieved by
chain-substrate incommensurabilities belonging to the class of non-quadratic
irrational numbers (e.g., the spiral mean). The well-known golden mean
(quadratic) incommensurability which slides best in the standard FK model shows
instead higher kinetic-friction values. The underlying reason lies in the
pinning properties of the lattice of solitons formed by the chain with the
substrate having the closest periodicity, with the other slider.Comment: 14 pagine latex - elsart, including 4 figures, submitted to Tribology
Internationa
A survey of some arithmetic applications of ergodic theory in negative curvature
This paper is a survey of some arithmetic applications of techniques in the
geometry and ergodic theory of negatively curved Riemannian manifolds, focusing
on the joint works of the authors. We describe Diophantine approximation
results of real numbers by quadratic irrational ones, and we discuss various
results on the equidistribution in , and in the
Heisenberg groups of arithmetically defined points. We explain how these
results are consequences of equidistribution and counting properties of common
perpendiculars between locally convex subsets in negatively curved orbifolds,
proven using dynamical and ergodic properties of their geodesic flows. This
exposition is based on lectures at the conference "Chaire Jean Morlet:
G\'eom\'etrie et syst\`emes dynamiques", at the CIRM, Luminy, 2014. We thank B.
Hasselblatt for his strong encouragements to write this survey.Comment: 31 pages, 15 figure
On the arithmetic of crossratios and generalised Mertens' formulas
We develop the relation between hyperbolic geometry and arithmetic
equidistribution problems that arises from the action of arithmetic groups on
real hyperbolic spaces, especially in dimension up to 5. We prove
generalisations of Mertens' formula for quadratic imaginary number fields and
definite quaternion algebras over the rational numbers, counting results of
quadratic irrationals with respect to two different natural complexities, and
counting results of representations of (algebraic) integers by binary
quadratic, Hermitian and Hamiltonian forms with error bounds. For each such
statement, we prove an equidistribution result of the corresponding
arithmetically defined points. Furthermore, we study the asymptotic properties
of crossratios of such points, and expand Pollicott's recent results on the
Schottky-Klein prime functions.Comment: 44 page
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