Dimension approximation in smooth dynamical systems

Abstract

For 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 ${\lambda }_u(\mu )>0>{\lambda }_s(\mu )$, 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 $\mu$.

23 pages