Gradual doubling property of Hutchinson orbits

Aimar, Hugo, Carena, Marilina, and Iaffei, Bibiana. "Gradual doubling property of Hutchinson orbits." Czechoslovak Mathematical Journal 65.1 (2015): 191-205. <http://eudml.org/doc/270055>.

@article{Aimar2015,
abstract = {The classical self-similar fractals can be obtained as fixed points of the iteration technique introduced by Hutchinson. The well known results of Mosco show that typically the limit fractal equipped with the invariant measure is a (normal) space of homogeneous type. But the doubling property along this iteration is generally not preserved even when the starting point, and of course the limit point, both have the doubling property. We prove that the elements of Hutchinson orbits possess the doubling property except perhaps for radii which decrease to zero as the step of the iteration grows, and in this sense, we say that the doubling property of the limit is achieved gradually. We use this result to prove the uniform upper doubling property of the orbits.},
author = {Aimar, Hugo, Carena, Marilina, Iaffei, Bibiana},
journal = {Czechoslovak Mathematical Journal},
keywords = {metric space; doubling measure; Hausdorff-Kantorovich metric; iterated function system},
language = {eng},
number = {1},
pages = {191-205},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Gradual doubling property of Hutchinson orbits},
url = {http://eudml.org/doc/270055},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Aimar, Hugo
AU - Carena, Marilina
AU - Iaffei, Bibiana
TI - Gradual doubling property of Hutchinson orbits
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 191
EP - 205
AB - The classical self-similar fractals can be obtained as fixed points of the iteration technique introduced by Hutchinson. The well known results of Mosco show that typically the limit fractal equipped with the invariant measure is a (normal) space of homogeneous type. But the doubling property along this iteration is generally not preserved even when the starting point, and of course the limit point, both have the doubling property. We prove that the elements of Hutchinson orbits possess the doubling property except perhaps for radii which decrease to zero as the step of the iteration grows, and in this sense, we say that the doubling property of the limit is achieved gradually. We use this result to prove the uniform upper doubling property of the orbits.
LA - eng
KW - metric space; doubling measure; Hausdorff-Kantorovich metric; iterated function system
UR - http://eudml.org/doc/270055
ER -