Equilibriums of some non-Hölder potentials

We consider one-sided subshifts with some potential functions which satisfy the Hölder condition everywhere except at a fixed point and its preimages. We prove that the systems have conformal measures and invariant measures absolutely continuous with respect to , where may be finite or infinite. We show that the systems are exact, and are weak Gibbs measures and equilibriums for . We also discuss uniqueness of equilibriums and phase transition.

These results can be applied to some expanding dynamical systems with an indifferent fixed point.