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We show that various possible versions of the Brjuno function, based on diierent kinds of continued fraction developments, are all equivalent and we study their regularity (L p , BMO and HH older) properties, through a systematic analysis of the functional equation which they fullll. Abstract We show that various possible versions of the Brjuno function,… (More)
Genetic low density lipoprotein (LDL) deficiency and high density lipoprotein (HDL) excess have been associated with enhanced longevity. This investigation assessed the prevalence of lipoprotein abnormalities in octogenarians free of clinical evidence of cardiovascular disease (CVD) in the Framingham Heart Study. Plasma lipid and lipoprotein cholesterol… (More)
We exhibit an explicit full measure class of minimal interval exchange maps T for which the cohomological equation Ψ − Ψ • T = Φ has a bounded solution Ψ provided that the datum Φ 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… (More)
The continued fraction expansion of the real number x = a0 + x0, a0 ∈ Z Z, is given by 0 ≤ xn < 1, x −1 n = an+1 + xn+1, an+1 ∈ IN, for n ≥ 0. The Brjuno function is then B(x) = ∞ n=0 x0x1. .. xn−1 ln(x −1 n), and the number x satisfies the Brjuno diophantine condition whenever B(x) is bounded. Invariant circles under a complex rotation persist when the map… (More)
The small divisors problem which is raised by stability questions in classical mechanics has an analog in holomorphic dynamical systems, namely the existence of Siegel disks. The size of these disks is well represented by the Brjuno functions. We analyse the relation between these functions and the various continued fraction transformations, and display the… (More)
Using thermodynamical formalism, we set up the perturbative approach for the Hausdorr dimension and the dimension spectrum of the Julia set associated with polynomials close to monomials. We extend previous calculations to a more general case where the perturbation is not constant.
For almost all interval exchange maps T 0 , with combinatorics of genus g ≥ 2, we construct affine interval exchange maps T which are semi–conjugate to T 0 and have a wandering interval. (maps of the interval), 11J70 (Continued fractions and generalizations) CONTENTS 0. Introduction 1. The continued fraction algorithm for interval exchange maps 1.1 Interval… (More)