Search

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]

An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to 1 is an eigenform for suitable weights on the cells. [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]

I give some rather general criteria for the uniqueness of the eigenform on finitely ramified fractals. In particular, I characterize the uniqueness in the case of four vertices with some additional condition, and give some examples of non-tree symmetric fractals where the eigenform is not unique. [ABSTRACT FROM AUTHOR]