Fractals

This group explored the relationship between limit spaces of contracting self-similar groups and self-similar structures. They gave the condition on a contracting group such that its limit space admits a self-similar structure, and also the condition such that this self-similar structure is p.c.f. They then gave the necessary and sufficient condition on a p.c.f. self-similar structure such that there exists a contracting group whose limit space has an isomorphic self-similar structure; in this case, they provided a construction that produces such a contracting group. Finally, they illustrated their results with several examples.

Publications

arXiv.org
International Journal of Algebra and Computation (IJAC)

Spectrum and Heat Kernel Asymptotics on General Laakso Spaces

Posted on April 14, 2016December 24, 2016 by

Contributors

Matthew Begue, Levi deValve, David Miller, and Benjamin Steinhurst

Overview

This group introduced a method of constructing a general Laakso space while calculating the spectrum and multiplicities of the Laplacian operator on it. Using this information, they found the leading term of the trace of the heat kernel of a Laakso space and its spectral dimension.

Publications

arXiv.org
Fractals

Modified Hanoi Towers Groups and Limit Spaces

Posted on April 14, 2016December 24, 2016 by

Contributors

Shotaro Makisumi, Grace Stadnyk, and Benjamin Steinhurst

Overview

This group introduced the k-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and gave conditions for them to be contractive. They analyzed the limit spaces of a particular family of contracting Hanoi groups, and showed that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, they provided partial results on the contraction of Hanoi groups with weaker symmetry.

Publications

arXiv.org
International Journal of Algebra and Computation (IJAC)

Eigenmodes of a Laplacian on Laakso Space

Posted on April 14, 2016December 24, 2016 by

Contributors

Kevin Romeo and Benjamin Steinhurst

Overview

This group analyzed the spectrum of a self-adjoint operator on a Laakso space using the projective limit construction originally given by Barlow and Evans. They used the hierarchical cell structure induced by the choice of approximating quantum graphs to calculate the spectrum with multiplicities. They then extended the method for using the hierarchical cell structure to more general projective limits beyond Laakso spaces.

Publications

Complex Variables and Elliptic Equations: An International Journal
arXiv.org

Vibration Spectra of the m-Tree Fractal

Posted on April 14, 2016December 24, 2016 by

Contributors

Daniel Ford and Benjamin Steinhurst

Overview

This group introduced a family of post-critically finite fractal trees indexed by the number of branches they possess. They then produced a Laplacian operator on graph approximations to these fractals and used spectral decimation to describe the spectrum of the Laplacian on these trees. Lastly they considered the behavior of the spectrum as the number of branches increases.