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.

中文翻译：

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非均匀双曲系统中的维数估计和近似

让 $f: M\右箭头 M$ 成为一个 $C^{1+\alpha }$ 上的微分同胚 $m_0$ 维紧致光滑黎曼流形中号和 $\亩$ 双曲遍历F-不变概率测度。本文获得了稳定（不稳定）点维数的上限 $\亩$ ，它由涉及次加性测量理论压力的方程的唯一解给出。如果 $\亩$ 是 Sinai-Ruelle-Bowen (SRB) 测度，那么 Kaplan-Yorke 猜想在一些附加条件下成立并且 Lyapunov 维数为 $\亩$ 可以通过双曲集序列的豪斯多夫维数逐渐逼近 $\{\Lambda _n\}_{n\geq 1}$ 。 Carathéodory 奇异维度的极限行为 $\Lambda _n$ 还研究了关于超加性奇异值势的不稳定流形。