Harmonic functions on the Sierpinski triangle

TL;DR

This paper investigates the local behavior of harmonic functions on the Sierpinski triangle, providing formulas for their Hölder exponents, an algorithm for their calculation at rational points, and analyzing their derivatives.

Contribution

It introduces a general formula for the Hölder exponent of harmonic functions on the Sierpinski triangle and an explicit algorithm for computing it at rational points.

Findings

Harmonic functions have derivatives that are always 0, infinity, or undefined.

A formula for the Hölder exponent at any point on the triangle's side.

An algorithm to compute the Hölder exponent at rational points.

Abstract

In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the H\"older exponent of such a function in a given point. From this formula, we deduce an explicit algorithm to calculate this exponent in any rational point, and the fact that the derivative of such a function is always equal to 0, infinity or undefined.