A p.c.f. self-similar set with no self-similar energy
Roberto Peirone
Università di Roma Tor Vergata, Italy
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Abstract
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.
Cite this article
Roberto Peirone, A p.c.f. self-similar set with no self-similar energy. J. Fractal Geom. 6 (2019), no. 4, pp. 393–404
DOI 10.4171/JFG/82