Singularity of self-similar measures with respect to Hausdorff measures
Besicoviteh (1941) and Egglestone (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-p expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focu...
| Authors: | ; |
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| Format: | Report |
| Publication date: | 1995 |
| Country: | España |
| Institution: | Universidad Complutense de Madrid (UCM) |
| Repository: | Docta Complutense |
| Language: | English |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/64093 |
| Online Access: | https://hdl.handle.net/20.500.14352/64093 |
| Access Level: | Open access |
| Keyword: | Self-similar measures Hausdorff measures Econometría (Economía) 5302 Econometría |
| Summary: | Besicoviteh (1941) and Egglestone (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-p expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the law of the iterated logarithm |
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