Let be a mapping from a metric space X to a metric space Y, and let α 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)) = αdim (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 = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for every E ⊆ M.
@article{urn:eudml:doc:41836,
title = {Linear distortion of Hausdorff dimension and Cantor's function.},
journal = {Collectanea Mathematica},
volume = {57},
year = {2006},
pages = {193-210},
zbl = {1093.28004},
mrnumber = {MR2223852},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41836}
}
Dovgoshey, Oleksiy; Ryazanov, Vladimir; Martio, Olli; Vuorinen, Matti. Linear distortion of Hausdorff dimension and Cantor's function.. Collectanea Mathematica, Tome 57 (2006) pp. 193-210. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41836/