  We use the concept of Baire Ergodicity and Ergodic Formalism introduced to
study topological and statistical attractors for interval maps, even with
discontinuities. For that we also analyze the {\em wandering intervals
attractors}. As a result, we establish the finiteness of the non-periodic
topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$
interval maps without discontinuities, we show the coincidence of the
statistical attractors with the topological ones and we calculate the upper
Birkhoff averages of continuous functions for generic points, even when the map
has abundance of historical behavior.
