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 Φ=(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.
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