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.

Peirone, R. (2019). A p.c.f. self-similar set with no self-similar energy. JOURNAL OF FRACTAL GEOMETRY, 6(4), 393-404 [10.4171/JFG/82].

A p.c.f. self-similar set with no self-similar energy

Peirone, Roberto
2019

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.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05
English
Self-similar energies; finitely ramified fractals; eigenforms
Peirone, R. (2019). A p.c.f. self-similar set with no self-similar energy. JOURNAL OF FRACTAL GEOMETRY, 6(4), 393-404 [10.4171/JFG/82].
Peirone, R
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/245591
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