1 results
Dimension estimates and approximation in non-uniformly hyperbolic systems
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 10 / October 2024
- Published online by Cambridge University Press:
- 12 February 2024, pp. 2975-3001
- Print publication:
- October 2024
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Let
$f: M\rightarrow M$ be a 
$C^{1+\alpha }$ diffeomorphism on an 
$m_0$-dimensional compact smooth Riemannian manifold M and 
$\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of 
$\mu $, which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If 
$\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of 
$\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets 
$\{\Lambda _n\}_{n\geq 1}$. The limit behaviour of the Carathéodory singular dimension of 
$\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.
