572 research outputs found
Diophantine conditions and real or complex Brjuno functions
The continued fraction expansion of the real number x=a_0+x_0, a_0\in
{\ZZ}, is given by 0\leq x_n<1, x_{n}^{-1}=a_{n+1}+ x_{n+1}, a_{n+1}\in
{\NN}, for The Brjuno function is then
and the number
satisfies the Brjuno diophantine condition whenever is bounded.
Invariant circles under a complex rotation persist when the map is analytically
perturbed, if and only if the rotation number satisfies the Brjuno condition,
and the same holds for invariant circles in the semi-standard and standard maps
cases. In this lecture, we will review some properties of the Brjuno function,
and give some generalisations related to familiar diophantine conditions. The
Brjuno function is highly singular and takes value on a dense set
including rationals. We present a regularisation leading to a complex function
holomorphic in the upper half plane. Its imaginary part tends to the Brjuno
function on the real axis, the real part remaining bounded, and we also
indicate its transformation under the modular group.Comment: latex jura.tex, 6 files, 19 pages Proceedings on `Noise, Oscillators
and Algebraic Randomness' La Chapelle des Bois, France 1999-04-05 1999-04-10
April 5-10, 1999 [SPhT-T99/116
Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
We consider the susceptibility function Psi(z) of a piecewise expanding
unimodal interval map f with unique acim mu, a perturbation X, and an
observable phi. Combining previous results (deduced from spectral properties of
Ruelle transfer operators) with recent work of Breuer-Simon (based on
techniques from the spectral theory of Jacobi matrices and a classical paper of
Agmon), we show that density of the postcritical orbit (a generic condition)
implies that Psi(z) has a strong natural boundary on the unit circle. The
Breuer-Simon method provides uncountably many candidates for the outer
functions of Psi(z), associated to precritical orbits. If the perturbation X is
horizontal, a generic condition (Birkhoff typicality of the postcritical orbit)
implies that the nontangential limit of the Psi(z) as z tends to 1 exists and
coincides with the derivative of the acim with respect to the map (linear
response formula). Applying the Wiener-Wintner theorem, we study the
singularity type of nontangential limits as z tends to e^{i\omega}. An
additional LIL typicality assumption on the postcritical orbit gives stronger
results.Comment: LaTex, 23 pages, to appear ETD
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
Linearization of analytic and non--analytic germs of diffeomorphisms of
We study Siegel's center problem on the linearization of germs of
diffeomorphisms in one variable. In addition of the classical problems of
formal and analytic linearization, we give sufficient conditions for the
linearization to belong to some algebras of ultradifferentiable germs closed
under composition and derivation, including Gevrey classes.
In the analytic case we give a positive answer to a question of J.-C. Yoccoz
on the optimality of the estimates obtained by the classical majorant series
method.
In the ultradifferentiable case we prove that the Brjuno condition is
sufficient for the linearization to belong to the same class of the germ. If
one allows the linearization to be less regular than the germ one finds new
arithmetical conditions, weaker than the Brjuno condition. We briefly discuss
the optimality of our results.Comment: AMS-Latex2e, 11 pages, in press Bulletin Societe Mathematique de
Franc
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
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