We introduce a set of hyperbolic and weakly-temperable points, called HWT. We then introduce a leaf condition- the existence of an unstable leaf for which the leaf volume of HWT is positive. We characterize the existence of a hyperbolic SRB measure by the leaf condition. We introduce the notion of GSRB- generalized SRB measures, and characterize their existence by a leaf condition. We show physicality of GSRB measures, their uniqueness on ergodic homoclinic classes via thermodynamic formalism arguments; and show that an ergodic component of a GSRB is a GSRB. In addition, we characterize the leaf condition by recurrence and 0 pressure of the geometric potential.
