Geodesic
On the Spectrum of Hecke Type Operators Related to Some Fractal Groups
Informatics and Automation, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 5-45

Voir la notice de l'article provenant de la source Math-Net.Ru

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also “substitutional graphs”. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined in \cite {grigorchuk:burnside}.\lb In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process. 
@article{TRSPY_2000_231_a0,
     author = {L. Bartholdi and R. I. Grigorchuk},
     title = {On the {Spectrum} of {Hecke} {Type} {Operators} {Related} to {Some} {Fractal} {Groups}},
     journal = {Informatics and Automation},
     pages = {5--45},
     publisher = {mathdoc},
     volume = {231},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2000_231_a0/}
}
TY  - JOUR
AU  - L. Bartholdi
AU  - R. I. Grigorchuk
TI  - On the Spectrum of Hecke Type Operators Related to Some Fractal Groups
JO  - Informatics and Automation
PY  - 2000
SP  - 5
EP  - 45
VL  - 231
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2000_231_a0/
LA  - en
ID  - TRSPY_2000_231_a0
ER  - 
%0 Journal Article
%A L. Bartholdi
%A R. I. Grigorchuk
%T On the Spectrum of Hecke Type Operators Related to Some Fractal Groups
%J Informatics and Automation
%D 2000
%P 5-45
%V 231
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2000_231_a0/
%G en
%F TRSPY_2000_231_a0
L. Bartholdi; R. I. Grigorchuk. On the Spectrum of Hecke Type Operators Related to Some Fractal Groups. Informatics and Automation, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 5-45. http://geodesic.mathdoc.fr/item/TRSPY_2000_231_a0/