Infinite dimensional i.f.s. and smooth functions on the Sierpinski gasket
2007 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, Vol. 56, no 3, 1377-1404 p.Article in journal (Refereed) Published
We describe the infinitesimal geometric behavior of a large class of intrinsically smooth functions on the Sierpinski gasket in terms of the limit distribution of their local eccentricity, which is essentially the direction of the gradient. The distribution of eccentricicies is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. Continuity properties of the gradient are used to define a class of nearly harmonic functions which are well approximated by harmonic functions. The gradient is also used to identify the part of the Sierpinski gasket where a smooth function is nearly harmonic locally. We prove that for nearly harmonic functions the limit distribution is the same as that for harmonic functions found by Oberg, Strichartz and Yingst. In particular, we prove convergence in the Wasserstein metric. We consider uniform as well as energy weights.
Place, publisher, year, edition, pages
2007. Vol. 56, no 3, 1377-1404 p.
IdentifiersURN: urn:nbn:se:uu:diva-95493ISI: 000247849000013OAI: oai:DiVA.org:uu-95493DiVA: diva2:169731