Linear distortion of Hausdorff dimension and Cantor's function

• Autores: Matti Vuorinen, V. Ryazanov
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 57, Fasc. 2, 2006, págs. 193-210
• Idioma: inglés
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• Resumen

  • Let f be a mapping from a metric space X to a metric space Y , and let ! be a positive real number. Write dim(E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim(f(E)) = ! dim(E) holds for each E " X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log 2/log 3 and dim(G(E)) = (log 3/log 2) dim(E), for every E " M.