Geodesic
Dimension of random fractals in metric spaces
Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 3, pp. 589-616

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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.
Mots-clés : Hausdorff dimension, local dimension, Markov chain, fractal.
Keywords: Hausdorff measure 
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     author = {A. A. Tempel'man},
     title = {Dimension of random fractals in metric spaces},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {589--616},
     publisher = {mathdoc},
     volume = {44},
     number = {3},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1999_44_3_a5/}
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A. A. Tempel'man. Dimension of random fractals in metric spaces. Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 3, pp. 589-616. http://geodesic.mathdoc.fr/item/TVP_1999_44_3_a5/