  The relation between saddle points of the potential of a classical
many-particle system and the analyticity properties of its thermodynamic
functions is studied. For finite systems, each saddle point is found to cause a
nonanalyticity in the Boltzmann entropy, and the functional form of this
nonanalytic term is derived. For large systems, the order of the nonanalytic
term increases unboundedly, leading to an increasing differentiability of the
entropy. Analyzing the contribution of the saddle points to the density of
states in the thermodynamic limit, our results provide an explanation of how,
and under which circumstances, saddle points of the potential energy landscape
may (or may not) be at the origin of a phase transition in the thermodynamic
limit. As an application, the puzzling observations by Risau-Gusman et al. on
topological signatures of the spherical model are elucidated.
