Spectra of non-commutative dynamical systems and graphs related to fractal groups

@article{Bartholdi2000SpectraON,
  title={Spectra of non-commutative dynamical systems and graphs related to fractal groups},
  author={Laurent Bartholdi and Rostislav I. Grigorchuk},
  journal={Comptes Rendus Mathematique},
  year={2000},
  volume={331},
  pages={429-434}
}
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13 Citations
Background Citations
4
Methods Citations
3

13 Citations

VIBRATION SPECTRA OF THE m-TREE FRACTAL
TLDR
A family of post-critically finite fractal trees indexed by the number of branches they possess is introduced and a Laplacian operator is produced on graph approximations to these fractals and spectral decimation is used to describe the spectrum of the LaplACian on these trees.
Spectral properties of graphs associated to the Basilica group
We provide the foundation of the spectral analysis of the Laplacian on the orbital Schreier graphs of the basilica group, the iterated monodromy group of the quadratic polynomial $z^2-1$. This group…
Integrable and Chaotic Systems Associated with Fractal Groups
TLDR
This paper provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, and contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.
The Prime Geodesic Theorem and Quantum Mechanics on Finite Volume Graphs
The prime geodesic theorem is reviewed for compact and finite volume Riemann surfaces and for finite and finite volume graphs. The methodology of how these results follow from the theory of the…
On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups
We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are…
Diffusions and Laplacians on Laakso, Barlow-Evans, and other fractals
We introduce a family of post-critically finite fractal trees indexed by the number of branches they possess. Then we produce a Laplacian operator on graph approximations to these fractals and use…
On a Torsion-Free Weakly Branch Group Defined by a Three State Automaton
We study a torsion free weakly branch group G without free subgroups defined by a three state automaton which appears in different problems related to amenability, Galois groups and monodromy. Here...
Renormalisation of random hierarchial systems
 This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call random hierarchical systems ) of probability…
Endomorphic presentations of branch groups
Solved and Unsolved Problems Around One Group
This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate…
...
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References

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Répartition asymptotique des valeurs propres de l’opérateur de Hecke _
La répartition asymptotique des valeurs propres des opérateurs de Hecke Tp, pour p premier variable, est un problème intéressant et difficile, sur lequel on ne dispose que de résultats partiels, cf.…
Symmetric random walks on groups
Zur allgemeinen Theorie des Masses
Cayley graphs: eigenvalues, expanders and random walks
Spectra of non-commutative dynamical systems and graphs related to fractal groups
  • Comptes rendus mathématique,
  • 2000