Dimension approximation in smooth dynamical systems

For a non-conformal repeller Λ of a C1+α map f preserving an ergodic measure μ of positive entropy, this paper shows that the Lyapunov dimension of μ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a C1+α diffeomorphism f preserving a hyperbolic ergodic measure μ of positive entropy, if (f, μ ) has only two Lyapunov exponents x3bb ᵤ(μ )>0>x3bb ₛ(μ ) , then the Hausdorff or lower box or upper box dimension of μ can be approximated by the corresponding dimension of the horseshoes \Λ ₙ\ . The same statement holds true if f is a C¹ diffeomorphism with a dominated Oseledet’s splitting with respect to μ .