284 research outputs found

    On cross-ratio distortion and Schwarz derivative

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    We prove asymptotic estimates for the cross-ratio distortion with respect to a smooth or holomorphic function in terms of its Schwarz derivative.Comment: the spelling of the name `Schwarz' correcte

    Herman's Theory Revisited

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    We prove that a C2+αC^{2+\alpha}-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class DδD_\delta, 0<δ<α≤10<\delta<\alpha\le1, is C1+α−δC^{1+\alpha-\delta}-smoothly conjugate to a rigid rotation. We also derive the most precise version of Denjoy's inequality for such diffeomorphisms.Comment: 10 page

    Holomorphic linearization of commuting germs of holomorphic maps

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    Let f1,...,fhf_1, ..., f_h be h≥2h\ge 2 germs of biholomorphisms of \C^n fixing the origin. We investigate the shape a (formal) simultaneous linearization of the given germs can have, and we prove that if f1,...,fhf_1, ..., f_h commute and their linear parts are almost simultaneously Jordanizable then they are simultaneously formally linearizable. We next introduce a simultaneous Brjuno-type condition and prove that, in case the linear terms of the germs are diagonalizable, if the germs commutes and our Brjuno-type condition holds, then they are holomorphically simultaneously linerizable. This answers to a multi-dimensional version of a problem raised by Moser.Comment: 24 pages; final version with erratum (My original paper failed to cite the work of L. Stolovitch [ArXiv:math/0506052v2]); J. Geom. Anal. 201

    Renormalisation scheme for vector fields on T2 with a diophantine frequency

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    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

    Local dynamics for fibered holomorphic transformations

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    Fibered holomorphic dynamics are skew-product transformations over an irrational rotation, whose fibers are holomorphic functions. In this paper we study such a dynamics on a neighborhood of an invariant curve. We obtain some results analogous to the results in the non fibered case

    Affine interval exchange maps with a wandering interval

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    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.Comment: 43 pages, 1 figur

    Rigidity of centralizers of diffeomorphisms

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    Centralizers of Anosov diffeomorphisms on tori

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    Universal Mandelbrot Set as a Model of Phase Transition Theory

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    The study of Mandelbrot Sets (MS) is a promising new approach to the phase transition theory. We suggest two improvements which drastically simplify the construction of MS. They could be used to modify the existing computer programs so that they start building MS properly not only for the simplest families. This allows us to add one more parameter to the base function of MS and demonstrate that this is not enough to make the phase diagram connectedComment: 5 pages, 3 figure
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