A Sierpinski carpet like fractal without standard self-similar energy

TL;DR

This paper constructs a fractal similar to the Sierpinski carpet that lacks a self-similar diffusion with sub-Gaussian heat kernel estimates, challenging previous assumptions about such diffusions on fractals.

Contribution

It introduces a new fractal example where self-similar diffusion with sub-Gaussian estimates does not exist, expanding understanding of diffusion behavior on fractals.

Findings

Constructs a Sierpinski carpet-like fractal without self-similar diffusion

Shows the absence of sub-Gaussian heat kernel estimates on this fractal

Contrasts with previous results on generalized Sierpinski carpets

Abstract

We construct a Sierpinski carpet like fractal, on which a self-similar diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets.