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 \(\Phi = (Q,P): {{\mathbb {R}}}^{d} \times {{\mathbb {R}}}^{d} \rightarrow {{\mathbb {R}}}^{d} \times {{\mathbb {R}}}^{d}\) on the variables (q, p) such that for every \(p\in {{\mathbb {R}}}^d\) the induced map \(q\mapsto Q(q,p)\) is a diffeomorphism of \({{\mathbb {R}}}^d\). 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.