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An example of a p.c.f. (post-critically finite) self-similar set without eigenform for any set of weights, is provided. The existence of an eigenform on such sets was an important, long-standing open problem in analysis on fractals. This problem is related to that of existence of self-similar energies on fractals, because self-similar energies on p.c.f. self-similar sets are obtained by using eigenforms (and only in this way). A general existence result of self-similar energies was previously established by the present author with respect to a weaker version of self-similarity. [ABSTRACT FROM AUTHOR]

In this article we study two problems about the existence of a distance d on a given fractal having certain properties. In the first problem, we require that the maps ψi defining the fractal be Lipschitz of prescribed constants less than 1 with respect to the distance d, and in the second one, we require that arbitrary compositions of the maps ψi be uniformly bi-Lipschitz of related constants. Both problems have been investigated previously by other authors. In this article, on a large class of finitely ramified fractals,we prove that these two problems are equivalent and give a necessary and sufficient condition for the existence of such a distance. Such a condition is expressed in terms of asymptotic behavior of the product of certain matrices associated to the fractal. [ABSTRACT FROM AUTHOR]

I give an explicitly verifiable necessary and sufficient condition for the uniqueness of the eigenform on finitely ramified fractals, once an eigenform is known. This improves the results of my previous paper [16], where I gave some necessary and some sufficient conditions, and with a relatively mild additional requirement on the known eigenform. The result of this paper can be interpreted as a uniqueness result for self-similar energies on finitely ramified fractals. [ABSTRACT FROM AUTHOR]