Fractal transformations of harmonic functions Michael F. Barnsley, Uta Freiberg Department of Mathematics, Australian National University, Canberra, ACT, Australia 28 December 2006 ABSTRACT The theory of fractal homeomorphisms is applied to transform a Sierpinski triangle into what we call a Kigami triangle. The latter is such that the corresponding harmonic functions and the corresponding Laplacian 4 take a relatively simple form. This provides an alternative approach to recent results of Teplyaev. Using a second fractal homeomorphism we prove that the outer boundary of the Kigami triangle possesses a continuous first derivative at every point. This paper shows that IFS theory and the chaos game algorithm provide important tools for analysis on fractals. 1 Introduction We begin this paper with a brief review of the recently developed theory of fractal tops and fractal transforma- tions. We emphasize that fractal transformations may be computed readily by means of the chaos game algorithm. Then we develop a beautiful application: we show how the theory may be applied to transform the Sierpinski triangle so that the corresponding harmonic functions and the corresponding Laplacian 4 take a relatively simple form. This provides an alternative approach to recent results of Teplyaev 31 28 29 30 . In particular, Kigami 16 17 appears to have written the first papers regarding representation of Sierpinski triangle in harmonic coordinates and Meyers 23 may have been to present the geometrical interpretation of the harmonic representation. Here we prove, by use the fractal homeomorphism theorem a second time and a third time, that the basic curves, from which the Kigami triangle may be constructed, are continuously differentiable. Other relevant references of which we are aware are Berger 9 , Kusuoka 18 19 20 , and also Strichartz 26 , top of page 194, where there is mention of three-by-three row-stochastic transformations in equations (5) and their relation to the corresponding two-by-two matrices in equations (8), with mention of the eigenvectors and eigenvalues of the latter. See also Barnsley 4 5 concerning the impedence functions and spectra of renormalizable electro-mechanical systems and their relation to Julia sets. 2 Hyperbolic IFS and Birkhoff ’s ergodic theorem Definition 1. Let (X,d X ) be a complete metric space. Let {f 1 ,f 2 , ..., f N } be a finite sequence of strictly 1