Fixed Points of anti-attracting maps and Eigenforms on Fractals

TL;DR

This paper presents a new fixed-point theorem approach to establish the existence of eigenforms, which are self-similar energies, on finitely ramified fractals, simplifying previous proofs and potentially enabling further improvements.

Contribution

It introduces a novel fixed-point theorem for anti-attracting maps to prove the existence of eigenforms on fractals, offering a shorter and more adaptable proof.

Findings

Provides a new fixed-point theorem for anti-attracting maps

Simplifies the proof of existence of eigenforms on fractals

Potentially enables further improvements in analysis on fractals

Abstract

An important problem in analysis on fractals is the existence of a self-similar energy on finitely ramified fractals. The self-similar energies are constructed in terms of eigenforms, that is, eigenvectors of a special nonlinear operator. Previous results by C. Sabot and V. Metz give conditions for the existence of an eigenform. In this paper, I give a different and probably shorter proof of the previous results, which appears to be suitable for improvements. Such a proof is based on a fixed-point theorem for anti-attracting maps on a convex set.