  We introduce the concepts of Baire Ergodicity and Ergodic Formalism,
employing them to study topological and statistical attractors. Specifically,
we establish the existence and finiteness of such attractors and provide
applications for maps of the interval, Viana maps, non-uniformly expanding
maps, partially hyperbolic systems, strongly transitive dynamics, and
skew-products.
  In a dynamical system with an abundance of historic behavior (encompassing
all systems with some hyperbolicity, particularly Axiom A systems), one can
show the existence of a residual set with zero measure for every invariant
probability measure. Hence, in principle, utilizing the classical ergodic
theory to control the asymptotic topological/statistical behavior of generic
orbits is not feasible.
  Nevertheless, the results presented here can also be applied to such a
system, contributing to the study of generic orbits in systems with an
abundance of historic behavior.
