Lyapunov Exponents, Entropy and Dimension Page: 29
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Where N(E) is the number of cubes of diameter c needed to cover X. If C(X) = C(X) then
this common called the box counting dimension denoted C(X).
One of the big advantages of using the box counting dimension is that it is quite easy to
calculate, both theoretically and experimentally. On the downside though it does not always
behave in the way one would like. There are countable sets for which the box counting
dimension is positive. A better notion is the Hausdorff dimension. It is constructed using
the Hausdorff measure, and fixes many of the problems found in capacity.
Definition 6.1.2. Let A be a subset of a compact metric space. The 6-mesh a-dimensional
Hausdorff measure is defined as follows:
Hb(A) in f{Z(diam(G)) : F is a ~ - mesh cover of A}
GEF
For each a, the 8-mesh Hausdorff measure is a metric outer measure. To define the
a-dimensional Hausdorff measure, take the limit as -- 0.
He (A) = lim (Hf (A))
Definition 6.1.3. The Hausdorff dimension, HD, of a set is given by:
HD(X) = inf {c : He = 0}
aL29
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Williams, Jeremy M. Lyapunov Exponents, Entropy and Dimension, thesis, August 2004; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc4559/m1/35/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .