Dimension estimates and approximation in non-uniformly hyperbolic systems

Let f: M→ M be a C1+α diffeomorphism on an m₀ -dimensional compact smooth Riemannian manifold M and μ a hyperbolic ergodic f -invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of μ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If μ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of μ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets \Λ ₙ≥ 1 . The limit behaviour of the Carathéodory singular dimension of Λ ₙ on the unstable manifold with respect to the super-additive singular valued potential is also studied.