# Linear distortion of Hausdorff dimension and Cantor's function.

Oleksiy Dovgoshey; Vladimir Ryazanov; Olli Martio; Matti Vuorinen

Collectanea Mathematica (2006)

- Volume: 57, Issue: 2, page 193-210
- ISSN: 0010-0757

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topDovgoshey, Oleksiy, et al. "Linear distortion of Hausdorff dimension and Cantor's function.." Collectanea Mathematica 57.2 (2006): 193-210. <http://eudml.org/doc/41836>.

@article{Dovgoshey2006,

abstract = {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.},

author = {Dovgoshey, Oleksiy, Ryazanov, Vladimir, Martio, Olli, Vuorinen, Matti},

journal = {Collectanea Mathematica},

keywords = {Teoría de la medida; Dimensión de Hausdorff; Conjuntos de Cantor; Cantor function; Cantor ternary set; Hausdorff dimension},

language = {eng},

number = {2},

pages = {193-210},

title = {Linear distortion of Hausdorff dimension and Cantor's function.},

url = {http://eudml.org/doc/41836},

volume = {57},

year = {2006},

}

TY - JOUR

AU - Dovgoshey, Oleksiy

AU - Ryazanov, Vladimir

AU - Martio, Olli

AU - Vuorinen, Matti

TI - Linear distortion of Hausdorff dimension and Cantor's function.

JO - Collectanea Mathematica

PY - 2006

VL - 57

IS - 2

SP - 193

EP - 210

AB - 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.

LA - eng

KW - Teoría de la medida; Dimensión de Hausdorff; Conjuntos de Cantor; Cantor function; Cantor ternary set; Hausdorff dimension

UR - http://eudml.org/doc/41836

ER -

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