Linear distortion of Hausdorff dimension and Cantor's function

: Dovgoshey O., Ryazanov V., Martio O., Vuorinen M.

: 2006

: Collectanea Mathematica

: 57

: 2

: 193

: 210

: 18

: 0010-0757

: http://api.elsevier.com/content/abstract/scopus_id:35348933857

Let f 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 H(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)) = α 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 H (M) = 1, with s = log 2/log 3 and dim(G(E)) = (log 3/log2) dim(E), for every E ⊆ M. © 2006 Universitat de Barcelona.