Fixed Points of anti-attracting maps and Eigenforms on Fractals

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.