Exact Dimensionality of Hyperbolic Measures

Abstract

In the survey article [17], Eckmann and Ruelle discussed various concepts of dimension and pointed out the importance of the so-called pointwise (local) dimension of invariant measures. For a Borel measure µ on a compact metric space M, the latter was introduced by Young in [95] and is defined by

$$d_\mu(x)\mathop= \limits^{{\rm def}} \mathop {\lim}\limits_{\rho\to 0} \frac{{\log \mu (B(x,\rho ))}}{{\log \rho }}$$

((X.1))

(provided the limit exists). As Barreira et al. indicated in [7], the notion is an important characteristic of the system. It plays a crucial role in dimension theory (see, for example, [19], the ICM address by Young [97, pp. 1232] and also [98, pp. 318]). In ergodic case the existence of the limit in (X.1) for a Borel probability measure µ on M implies the crucial fact that virtually all the known characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. The common value is called the fractal dimension of µ.