Self-similar Dirichlet forms on polygon carpets

TL;DR

This paper constructs symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on polygon carpets, generalizing Sierpinski carpets by allowing different intersection and contraction properties.

Contribution

It introduces two new classes of polygon carpets, perfect and bordered, and develops self-similar diffusions with heat kernel estimates on these structures.

Findings

Constructed symmetric self-similar diffusions on polygon carpets.

Established sub-Gaussian heat kernel estimates for these diffusions.

Generalized Sierpinski carpets to broader polygon-based fractals.

Abstract

We construct symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on two types of polygon carpets, which are natural generalizations of planner Sierpinski carpets (SC). The first ones are called perfect polygon carpets that are natural analogs of SC in that any intersection cells are either side-to-side or point-to-point. The second ones are called bordered polygon carpets which satisfy the boundary including condition as SC but allow distinct contraction ratios.