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We study ergodic optimization. For any diffeomorphism, measures with maximum total exponent can be defined. In order to define this notion, we begin with introducing the notation. Let M be a compact smooth Riemannian manifold without boundary. It is also assumed to be connected. Let T be a C-diffeomorphism. Consider a compact T -invariant set Λ. We denote by M (T,Λ) the space of all T -invariant Borel probability measures supported on Λ equipped with the weak-∗ topology. We say that Λ is a basic set of T if Λ is isolated and hyperbolic for T and T |Λ : Λ→ Λ is topologically transitive. We recall that Λ is isolated for T if there exists an open neighborhood U of Λ such that Λ = ⋂ i∈Z T (U) holds. Let DT (x) be the derivative of T at x ∈ M . We denote by J(T )(x) the Jacobian of T at x ∈ M , that is, the absolute value of the determinant of DT (x). In ergodic optimization, invariant Borel probability measures maximizing the integral of a given function are mainly considered. In our study, those of a specific function are investigated. Now we define measures with maximum total exponent.

MEASURES WITH MAXIMUM TOTAL EXPONENT OF <i>C</i><sup>1</sup>-DIFFEOMORPHISMS WITH BASIC SETS