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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Poincaré-Birkhoff Theorems in random dynamics
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by Álvaro Pelayo and Fraydoun Rezakhanlou PDF
Trans. Amer. Math. Soc. 370 (2018), 601-639 Request permission

Abstract:

We propose a generalization of the Poincaré-Birkhoff Theorem on area-preserving twist maps to area-preserving twist maps $F$ that are random with respect to an ergodic probability measure. In this direction, we will prove several theorems concerning existence, density, and type of the fixed points. To this end first we introduce a randomized version of generalized generating functions, and verify the correspondence between its critical points and the fixed points of $F$, a fact which we successively exploit in order to prove the theorems. The study we carry out needs to combine probabilistic techniques with methods from nonlinear PDE, and from differential geometry, notably Moser’s method and Conley-Zehnder theory. Our stochastic model in the periodic case coincides with the classical setting of the Poincaré-Birkhoff Theorem.
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Additional Information
  • Álvaro Pelayo
  • Affiliation: Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, California 92093-0112
  • MR Author ID: 731609
  • Email: alpelayo@math.ucsd.edu
  • Fraydoun Rezakhanlou
  • Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840
  • MR Author ID: 253698
  • Email: rezakhan@math.berkeley.edu
  • Received by editor(s): February 5, 2015
  • Received by editor(s) in revised form: March 24, 2016, and April 14, 2016
  • Published electronically: August 15, 2017
  • Additional Notes: The first author was supported by NSF Grants DMS-1055897, DMS-1518420, and DMS-0635607, a J. Tinsely Oden Faculty Fellowship from the University of Texas, and an Oberwolfach Leibniz Fellowship
    The second author was supported in part by NSF Grant DMS-1106526 and DMS-1407723

  • Dedicated: To Alan Weinstein on his 70th birthday, with admiration.
    The authors dedicate this article to Alan Weinstein, whose fundamental and deep insights in so many areas of geometry are a continuous source of inspiration.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 601-639
  • MSC (2010): Primary 60D05
  • DOI: https://doi.org/10.1090/tran/6967
  • MathSciNet review: 3717991