Totally dissipative measures for the shift and conformal σ-finite measures for the stable holonomies

Abstract : In this paper we investigate some results of ergodic theory with infinite measures for a subshift of finite type. We give an explicit way to construct σ-finite measures which are quasi-invariant by the stable holonomy and equivalent to the conditional measures of some σ-invariant measure. These σ-invariant measures are totally dissipative, σ-finite but satisfy a Birkhoff Ergodic-like Theorem. The constructions are done for the symbolic case, but can be extended for uniformly hyperbolic flows or diffeomorphisms.