The Random Arnold Conjecture: A New Probabilistic Conley-Zehnder Theory for Symplectic Maps

Abstract

Inspired by the classical Conley-Zehnder Theorem and the Arnold Conjecture in symplectic topology, we prove a number of probabilistic theorems about the existence and density of fixed points of symplectic strand diffeomorphisms in dimensions greater than 2. These are symplectic diffeomorphisms = (Q, P) : Rd × Rd → Rd×Rd on the variables(q, p)such that for every p ∈ Rd the induced map q → Q(q, p) is a diffeomorphism of Rd . In particular we verify that quasiperiodic symplectic strand diffeomorphisms have infinitely many fixed points almost surely, provided certain natural conditions hold (inspired by the conditions in the Conley-Zehnder Theorem). The paper contains also a number of theorems which go well beyond the quasiperiodic case. Overall the paper falls within the area of stochastic dynamics but with a very strong symplectic geometric motivation, and as such its main inspiration can be traced back to Poincaré’s fundamental work on celestial mechanics and the restricted 3-body problem.