Ergodic optimization restricted on certain subsets of invariant measures

Abstract

In this article, we pay attention to transitive dynamical systems having the shadowing property and the entropy functions are upper semi-continuous. As for these dynamical systems, when we consider ergodic optimization restricted on the subset of invariant measures whose metric entropy are equal to or greater than a given constant, we prove that for C0-generic real continuous functions, the ergodic optimization measure is unique, ergodic, has full support and has metric entropy equal to the given constant. Similar results also hold for suspension flows over transitive subshifts of finite type, Cr(r≥2)-generic geometric Lorenz attractors and C1-generic singular hyperbolic attractors.