The celebrated Birkhoff's ergodic theorem asserts that from a probabilistic viewpoint the times averages of "almost all" points converge to a space average. Motivated by the application of iterated function systems (IFS) to model central dynamics of partially hyperbolic diffeomorphisms, we will describe mild conditions that ensure that Birkhoff non-typical points form a Baire generic subset. If time permits we will provide some applicationsof this result in a partial hyperbolicity context. This is a ongoing joint work with my postdoctoral student G. Ferreira (UFMA).