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This article is devoted to the study of the historic set, which wasintroduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolicsystems via Pesin theory. Particularly, we give a conditional variationalprinciple for historic sets. Our results can be applied (i) to thediffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolicdiffeomorphisms described by Katok and several other classes of diffeomorphismsderived from Anosov systems.

Conditional Variational Principle for Historic Set in Some Nonuniformly Hyperbolic Systems