Abstract:
We study the local and Hausdorff dimensions of measures in function and sequence spaces and the Hausdorff dimension of such spaces with respect to deterministic and random ‘`scale metrics.’ Following ideas due to Billingsley and Furstenberg we show that the local dimension of a properly chosen probability measure is an efficient tool for the calculation of the Hausdorff dimension. In particular, the calculation of the Hausdorff dimension of a sequence space with respect to a deterministic scale metric with finite memory is reduced to the calculation of the local dimension of the associated Markov chain that can be found easily; both dimensions coincide with the solution of the generalized Moran equation specified by the scale metric. When the scale metric is random we come to a stochastic analogue of the Moran equation. These results are used as a ‘`leading special case’ in the study of the Hausdorff dimension of deterministic and random fractals in general metric spaces.
Keywords:Hausdorff dimension, Hausdorff measure, local dimension, Markov chain, fractal.