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Poincaré-Birkhoff Theorems in random dynamics

Authors: Álvaro Pelayo and Fraydoun Rezakhanlou
Journal: Trans. Amer. Math. Soc. 370 (2018), 601-639
MSC (2010): Primary 60D05
Published electronically: August 15, 2017
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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

Fraydoun Rezakhanlou
Affiliation: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840

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.
Article copyright: © Copyright 2017 American Mathematical Society

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