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

TL;DR

This paper pioneers a probabilistic approach to Conley-Zehnder Theory for symplectic maps, establishing the almost sure existence of infinitely many fixed points in certain high-dimensional cases.

Contribution

It introduces the first probabilistic theorems for fixed points of symplectic twist maps, extending classical theory into a new stochastic framework.

Findings

Quasiperiodic symplectic twist maps have infinitely many fixed points almost surely.

First probabilistic theorems about fixed point density in higher dimensions.

Results extend beyond quasiperiodic cases to more general symplectic maps.

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.