Fundamental domain of invariant sets and applications

Let $X$ be a compact metric space, $f:X\to X$ a homeomorphism and $\phi \in C(X,\mathbb {R})$ . We construct a fundamental domain for the set of points with finite peaks with respect to the induced cocycle $\{\phi _n\}$ . As applications, we give sufficient conditions for the transitive set of a non-conservative partially hyperbolic diffeomorphism to have positive Lebesgue measure, i.e., for an accessible partially hyperbolic diffeomorphism, if the set of points with finite peaks for the Jacobian cocycle is not of full volume, then the set of transitive points is of positive volume.