Dimension approximation in smooth dynamical systems

Abstract

AbstractFor a non-conformal repeller $\Lambda $ of a $C^{1+\alpha }$ map f preserving an ergodic measure $\mu $ of positive entropy, this paper shows that the Lyapunov dimension of $\mu $ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+\alpha }$ diffeomorphism f preserving a hyperbolic ergodic measure $\mu $ of positive entropy, if $(f, \mu )$ has only two Lyapunov exponents $\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$ , then the Hausdorff or lower box or upper box dimension of $\mu $ can be approximated by the corresponding dimension of the horseshoes $\{\Lambda _n\}$ . The same statement holds true if f is a $C^1$ diffeomorphism with a dominated Oseledet’s splitting with respect to $\mu $ .