Conditional Variational Principle for Historic Set in Some Nonuniformly Hyperbolic Systems

This article is devoted to the study of the historic set, which was introduced by Ruelle, of Birkhoff averges in some nonuniformly hyperbolic systems via Pesin theory. Particularly, we give a conditional variational principle for historic sets. Our results can be applied (i) to the diffeomorphisms on surfaces, (ii) to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems.