Geometry of dynamics, Lyapunov exponents, and phase transitions

The Hamiltonian dynamics of the classical planar Heisenberg model is numerically investigated in two and three dimensions. In three dimensions peculiar behaviors are found in the temperature dependence of the largest Lyapunov exponent and of other observables related to the geometrization of the dynamics. On the basis of a heuristic argument it is conjectured that the phase transition might correspond to a change in the topology of the manifolds whose geodesics are the motions of the system.