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Small denominators. I. Mapping of the circumference onto itself

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Collected Works

Part of the book series: Vladimir I. Arnold - Collected Works ((ARNOLD,volume 1))

Abstract

In the first part of the paper it is shown that analytic mappings of the circumference, differing little from a rotation, whose rotation number is irrational and satisfies certain arithmetical requirements, may be carried into a rotation by an analytic substitution of variables. In the second part we consider the space of mappings of the circumference onto itself and the place occupied in this space by mappings of various types. We indicate applications to the investigation of trajectories on the torus and to the Dirichlet problem for the equation of the string.

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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of California, 94720-3840, Berkeley, CA, USA

    Alexander B. Givental

  2. Department of Mathematics, University of Toronto, M5S 3G3, Toronto, ON, Canada

    Boris A. Khesin

  3. Control & Dynamical Systems and Applied & Computational Mathematics, California Institute of Technology, 91125-8100, Pasadena, CA, USA

    Jerrold E. Marsden

  4. Department of Mathematics, University of North Carolina, 27599-3250, Chapel Hill, NC, USA

    Alexander N. Varchenko

  5. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, 119991, Moscow, Russian Federation

    Oleg Ya. Viro

  6. Institute of Mathematical Sciences, Stony Brook University, 11794-3660, Stony Brook, NY, USA

    Vladimir M. Zakalyukin

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(2009). Small denominators. I. Mapping of the circumference onto itself. In: Givental, A., et al. Collected Works. Vladimir I. Arnold - Collected Works, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01742-1_10

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