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It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$) either is empty or carries full topological entropy (or pressure, see [

In this article we obtain a variational principle for saturated sets for maps with some non-uniform specification properties. More precisely, we prove that the topological entropy of saturated sets coincides with the smallest measure theoretical entropy among the invariant measures in the accumulation set. Using this fact we provide lower bounds for the topological entropy of the irregular set and the level sets in the multifractal analysis of Birkhoff averages for continuous observables. The topological entropy estimates use as tool a non-uniform specification property on topologically large sets, which we prove to hold for open classes of non-uniformly expanding maps. In particular we prove some multifractal analysis results for *C*^{1}-open classes of non-uniformly expanding local diffeomorphisms and Viana maps [

Consequently, we obtain that Pesin's entropy formula always holds for (1) volume-preserving

*Anosov*diffeomorphisms, (2) volume-preserving

*partially hyperbolic*diffeomorphisms with

*one-dimensional*center bundle, (3) volume-preserving diffeomorphisms

*far away from homoclinic tangency*, and (4)

*generic*volume-preserving diffeomorphisms.

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