Article contents
Dimension approximation in smooth dynamical systems
Published online by Cambridge University Press: 11 April 2023
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
$\unicode{x3bb} _u(\mu )>0>\unicode{x3bb} _s(\mu )$, 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 $.
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- Original Article
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- © The Author(s), 2023. Published by Cambridge University Press
References
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