Title: Infinite dimensional i.f.s. and smooth functions on the Sierpinski gasket

Authors: Anders Pelander and Alexander Teplyaev

Issue: Volume 56 (2007), Issue 3, 1377-1404

Abstract: We describe the infinitesimal geometric behavior of a large class of intrinsically smooth functions on the Sierpi\'nski gasket in terms of the limit distribution of their local \emph{eccentricity}, which is essentially the direction of the gradient. The distribution of eccentricities 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 \emph{nearly harmonic} functions which are well approximated by harmonic functions. The gradient is also used to identify the part of the Sierpi\'nski 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.