Self-similar Dirichlet form on pillow-type carpets--a short analytic construction
Shiping Cao, Hua Qiu, Yizhou Wang
TL;DR
The paper addresses the existence of self-similar Dirichlet forms on pillow-type carpets, a class of infinitely ramified fractals including the Sierpiński carpet, by developing a purely analytic, graph-approximation framework. It builds a compatible sequence of energies on graph levels $(V_n)$ and passes to an inverse-limit fractal $(F,d_F)$, establishing a trace-resistance theory and a decimation procedure that yield a limiting energy $\widetilde{\mathcal E}$ on $F$. A key achievement is the two-sided multiplicative control of resistance constants $\mathscr R_n$, together with a construction of a limiting, regular, strongly local Dirichlet form that is self-similar with respect to the pillow maps $\Psi_w$. The results provide a robust analytic toolkit for analysis on pillow-type carpets, extending techniques beyond p.c.f. fractals to non-p.c.f. settings like the Sierpiński carpet. The framework also clarifies the role of planarity and symmetry in enabling analytic control of energy across scales.
Abstract
We give a short, self-contained analytic proof of the existence of self-similar Dirichlet forms on pillow-type carpets, a family of infinitely ramified fractals that includes the Sierpiński carpet.