Equality of Pressures for Diffeomorphisms Preserving Hyperbolic Measures

For a uniformly hyperbolic diffeomorphism f , the induced volume deformation φu in the unstable subbundle over a compact f -invariant set significantly characterizes the geometry of the set as well as the dynamics in its neighborhood. Under the hypothesis of uniform hyperbolicity, and particularly for hyperbolic surface diffeomorphisms, a large number of dynamical quantifiers such as, for example, fractal dimensions, Lyapunov exponents, and escape rates, are captured through the topological pressure of φu. If f : M → M is a C1+ε diffeomorphism on a Riemannian manifold M (we assume that some Riemannian metric on M is fixed) and Λ is an f -invariant locally maximal set such that f |Λ is uniformly hyperbolic and satisfies specification (and hence is mixing), then the topological pressure Pf |Λ of the function φu(x) = − log |Jac dfx|Eu x | can be calculated through