Dirichlet forms on self-similar sets with overlaps

TL;DR

This paper introduces a new framework for analyzing Dirichlet forms and Laplacians on self-similar sets with overlaps, extending Kigami's techniques beyond p.c.f. sets to more complex structures.

Contribution

It defines the 'finitely ramified of finite type' (f.r.f.t.) nested structure, enabling analysis on a broader class of self-similar sets with overlaps.

Findings

Developed a graph-directed construction for f.r.f.t. sets.

Extended Kigami's analysis techniques to f.r.f.t. sets.

Provided examples with explicit Dirichlet forms.

Abstract

We study Dirichlet forms and Laplacians on self-similar sets with overlaps. A notion of "finitely ramified of finite type($f.r.f.t.$) nested structure" for self-similar sets is introduced. It allows us to reconstruct a class of self-similar sets in a graph-directed manner by a modified setup of Mauldin and Williams, which satisfies the property of finite ramification. This makes it possible to extend the technique developed by Kigami for analysis on $p.c.f.$ self-similar sets to this more general framework. Some basic properties related to $f.r.f.t.$ nested structures are investigated. Several non-trivial examples and their Dirichlet forms are provided.