Critical dynamics and trees (Q1113510)

We show that the dependence of the critical perturbation of Hamiltonian systems on the frequency v is related to the statistics of large planar trees, using a complex map for illustration. A critical function K(v) is given by the radii of convergence of power series whose coefficients are represented as sums over trees. A preliminary approximation based on this relation leads to an expression for the critical function as the solution of a transcendental equation, which appears to provide a reasonable approximation to the true function, and to have the right structure near the dense set of singularities.