Exercise 9

The Cantor set in the interval [0,1] is constructed by removing the interval ]1/3,2/3[ leaving the two intervals [0,1/3] and [2/3,1]. Frome these the middle third is removed and so on. Assume that the Cantor set is the active set for turbulence.

What is the probability, P_l, of finding an active region at the level l. What is the fractal dimension of the Cantor set.

Using the $\beta$-model, calculate the anomalous scaling exponents, $\zeta_p$ in this case.