Pointwise Dimension

$$D_P({\bf x}) \equiv \lim_{\epsilon\to 0} {\ln \mu(B_\epsilon({\bf x}))\over \ln\epsilon},$$

where $B_\epsilon({\bf x})$ is an $n$-D Ball of Radius $\epsilon$ centered at ${\bf x}$ and $\mu$ is the Probability Measure.

See also Ball, Probability Measure

References

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 541-545, 1995.