  We construct spectral triples for path spaces of stationary Bratteli diagrams
and study their associated mathematical objects, in particular their zeta
function, their heat kernel expansion and their Dirichlet forms. One of the
main difficulties to properly define a Dirichlet form concerns its domain. We
address this question in particular in the context of Pisot substitution tiling
spaces for which we find two types of Dirichlet forms: one of transversal type,
and one of longitudinal type. Here the eigenfunctions under the translation
action can serve as a good core for a non-trivial Dirichlet form. We find that
the infinitesimal generators can be interpreted as elliptic differential
operators on the maximal equicontinuous factor of the tiling dynamical system.
