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Exact Dimensionality of Hyperbolic Measures

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1978)

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.

Keywords

  • Lyapunov Exponent
  • Invariant Measure
  • Hausdorff Dimension
  • Borel Measure
  • Conditional Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2009 Springer-Verlag Berlin Heidelberg

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QUIAN, M., XIE, JS., ZHU, S. (2009). Exact Dimensionality of Hyperbolic Measures. In: Smooth Ergodic Theory for Endomorphisms. Lecture Notes in Mathematics(), vol 1978. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01954-8_10

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