## Hausdorff Dimension

Let be a Subset of a Metric Space . Then the Hausdorff dimension of is the Infimum of such that the -dimensional Hausdorff Measure of is 0. Note that this need not be an Integer.

In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with Fractal Perimeter in Lorentz's conjecture. However, in general, the proper dimension to use turns out to be the Minkowski-Bouligand Dimension (Schroeder 1991).

References

Federer, H. Geometric Measure Theory. New York: Springer-Verlag, 1969.

Hausdorff, F. ``Dimension und äußeres Maß.'' Math. Ann. 79, 157-179, 1919.

Ott, E. ``Appendix: Hausdorff Dimension.'' Chaos in Dynamical Systems. New York: Cambridge University Press, pp. 100-103, 1993.

Schroeder, M. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman, pp. 41-45, 1991.