The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps

2023/06/27 by Álvaro Pelayo, Pelayo, Álvaro, Fraydoun Rezakhanlou +1 · 1 citation
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Probability (math.PR) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2306.15586

openalex publication_date 2023/06/27 · openalex created_date 2023/06/29 · openalex updated_date 2026/07/28

Abstract

We take the first steps to develop Conley-Zehnder Theory, as conjectured by Arnold, in the world of probability. As far as we know, this paper provides the first probabilistic theorems about the density of fixed points of symplectic twist maps in dimensions greater than 2. In particular we will show that, when the analogue conditions to classical Conley-Zehnder theory hold, quasiperiodic symplectic twist maps have infinitely many fixed points almost surely. The paper contains also a number of theorems which go well beyond the quasiperiodic case.

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