Convergence of energy forms on Sierpinski gaskets with added rotated triangle

TL;DR

This paper investigates the convergence of resistance metrics and forms on a sequence of fractal spaces, specifically Sierpinski gaskets with added rotated triangles, focusing on the existence and uniqueness of self-similar Dirichlet forms.

Contribution

It introduces a method to define Dirichlet forms on non-p.c.f. fractals via $ ext{Gamma}$-convergence, extending analysis to more complex fractal structures.

Findings

Resistance metrics converge on the sequence of spaces

Existence and uniqueness of self-similar Dirichlet forms established

Dirichlet forms on non-p.c.f. fractals constructed as limits

Abstract

We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated triangles. The fractals depend on a parameter in a continuous way. When the parameter is irrational, the fractal is not post critically finite (p.c.f.), and there are infinitely many ways that two cells intersect. In this case, we will define the Dirichlet form as a limit in some $\Gamma$-convergence sense of the Dirichlet forms on p.c.f. fractals that approximate it.