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We consider a Cantor-like set as a geometric projection of a Ber- noulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) intro- duced dimension-like indices in the probability space of a stochastic process. Under suitable regularity conditions we nd closed formulae linking the Haus- dor, box and packing metric dimensions of the subsets of the Cantor{like set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of well-known fractal spaces boils down to computing dimensions in the unit interval endowed with a suitable metric. We use these results to generalize density theorems in Cantor{like spaces. We also give some examples to illustrate the application of our results.

The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces