Eckmann–Ruelle conjecture on pointwise dimensions of hyperbolic measures

Let $M$ be a compact smooth Riemannian manifold, let $f:M\to M$ be a $C^{1+\alpha}$ diffeomorphism, and let $\mu$ be a hyperbolic measure. The pointwise dimension of $\mu$ is the limit, when it exists, of

$$\lim_{r\to 0}\frac{\log\mu(B(x,r))}{\log r}.$$

Eckmann–Ruelle conjecture. For any hyperbolic measure $\mu$ of a $C^{1+\alpha}$ diffeomorphism $f$, the pointwise dimension exists almost everywhere and is constant.

The conjecture concerns whether hyperbolicity alone guarantees almost-everywhere existence and almost-everywhere constancy of pointwise dimension, extending the established relationships between entropy, Lyapunov exponents, and dimensions for hyperbolic measures. Its resolution status is not specified in the source.