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arXiv · 2301.06233

Dimension approximation in smooth dynamical systems

Yongluo Cao·Juan Wang·Yun Zhao
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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 Carath\'{e}odory 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 $\mu$.

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Yongluo Cao, Juan Wang, Yun Zhao. 2023-01-16. Dimension approximation in smooth dynamical systems. https://arxiv.org/abs/2301.06233

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