Electrical resistance of N-gasket fractal networks

Brighid Boyle, Kristin Cekala, David Ferrone, Neil Rifkin and Alexander Teplyaev

Vol. 233 (2007), No. 1, 15–40

DOI: 10.2140/pjm.2007.233.15

Abstract

We study self-similar local regular Dirichlet, or energy, forms on a class of fractal N-gaskets, which are generalizations of polygaskets. This is directly related to self-similar diffusions and resistor networks (electrical circuits). We prove existence and uniqueness, and also obtain explicit formulas for scaling factors and resistances (transition probabilities). We also study asymptotic behavior of these quantiles as the number of “sides” N of an N-gasket tends to infinity.

Keywords

self-similar Dirichlet form, energy form, fractal, polygasket, N-gasket, resistor network, electrical circuit

Mathematical Subject Classification 2000

Primary: 28A80

Secondary: 94C99, 60J45, 34B45, 31C25

Milestones

Received: 18 August 2006

Revised: 25 May 2007

Accepted: 5 June 2007

Published: 1 November 2007