115 research outputs found
Reducibility of cocycles under a Brjuno-R\"ussmann arithmetical condition
The arithmetics of the frequency and of the rotation number play a
fundamental role in the study of reducibility of analytic quasi-periodic
cocycles which are sufficiently close to a constant. In this paper we show how
to generalize previous works by L.H.Eliasson which deal with the diophantine
case so as to implement a Brjuno-Russmann arithmetical condition both on the
frequency and on the rotation number. Our approach adapts the Poschel-Russmann
KAM method, which was previously used in the problem of linearization of vector
fields, to the problem of reducing cocycles
A quasianalyticity property for monogenic solutions of small divisor problems
We discuss the quasianalytic properties of various spaces of functions
suitable for one-dimensional small divisor problems. These spaces are formed of
functions C^1-holomorphic on certain compact sets K_j of the Riemann sphere (in
the Whitney sense), as is the solution of a linear or non-linear small divisor
problem when viewed as a function of the multiplier (the intersection of K_j
with the unit circle is defined by a Diophantine-type condition, so as to avoid
the divergence caused by roots of unity). It turns out that a kind of
generalized analytic continuation through the unit circle is possible under
suitable conditions on the K_j's
Bounded type interval exchange maps
Irrational numbers of bounded type have several equivalent characterizations.
They have bounded partial quotients in terms of arithmetic characterization and
in the dynamics of the circle rotation, the rescaled recurrence time to
-ball of the initial point is bounded below. In this paper, we consider how
the bounded type condition of irrational is generalized into interval exchange
maps.Comment: 12 page
Potts models on hierarchical lattices and Renormalization Group dynamics
We prove that the generator of the renormalization group of Potts models on
hierarchical lattices can be represented by a rational map acting on a
finite-dimensional product of complex projective spaces. In this framework we
can also consider models with an applied external magnetic field and
multiple-spin interactions. We use recent results regarding iteration of
rational maps in several complex variables to show that, for some class of
hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of
the renormalization map.Comment: 21 pages, 7 figures; v3 revised, some issues correcte
Coupling the Yoccoz-Birkeland population model with price dynamics: chaotic livestock commodities market cycles
We propose a new model for the time evolution of livestock commodities which
exhibits endogenous deterministic stochastic behaviour. The model is based on
the Yoccoz-Birkeland integral equation, a model first developed for studying
the time-evolution of single species with high average fertility, a relatively
short mating season and density dependent reproduction rates. This equation is
then coupled with a differential equation describing the price of a livestock
commodity driven by the unbalance between its demand and supply. At its birth
the cattle population is split into two parts: reproducing females and cattle
for butchery. The relative amount of the two is determined by the spot price of
the meat. We prove the existence of an attractor and we investigate numerically
its properties: the strange attractor existing for the original
Yoccoz-Birkeland model is persistent but its chaotic behaviour depends also
from the price evolution in an essential way.Comment: 26 pages, 19 figure
The cohomological equation for Roth type interval exchange maps
We exhibit an explicit full measure class of minimal interval exchange maps T
for which the cohomological equation has a bounded
solution provided that the datum belongs to a finite codimension
subspace of the space of functions having on each interval a derivative of
bounded variation. The class of interval exchange maps is characterized in
terms of a diophantine condition of ``Roth type'' imposed to an acceleration of
the Rauzy--Veech--Zorich continued fraction expansion associated to T.
CONTENTS 0. Introduction 1. The continued fraction algorithm for interval
exchange maps 1.1 Interval exchnge maps 1.2 The continued fraction algorithm
1.3 Roth type interval exchange maps 2. The cohomological equation 2.1 The
theorem of Gottschalk and Hedlund 2.2 Special Birkhoff sums 2.3 Estimates for
functions of bounded variation 2.4 Primitives of functions of bounded variation
3. Suspensions of interval exchange maps 3.1 Suspension data 3.2 Construction
of a Riemann surface 3.3 Compactification of 3.4 The cohomological
equation for higher smoothness 4. Proof of full measure for Roth type 4.1 The
basic operation of the algorithm for suspensions 4.2 The Teichm\"uller flow 4.3
The absolutely continuous invariant measure 4.4 Integrability of 4.5 Conditions (b) and (c) have full measure 4.6 The main step
4.7 Condition (a) has full measure 4.8 Proof of the Proposition Appendix A
Roth--type conditions in a concrete family of i.e.m. Appendix B A non--uniquely
ergodic i.e.m. satsfying condition (a) ReferencesComment: 64 pages, 4 figures (jpeg files
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