Projects: Fractals

Quantum Mechanics on Laakso Spaces

April 20, 2016

Contributors

Christopher Kauffman, Robert Kesler, Amanda Parshall, Evelyn Stamey, and Benjamin Steinhurst

Overview

This group first reviewed the spectrum of the Laplacian operator on a general Laakso space before considering modified Hamiltonians for the infinite square well, parabola, and Coulomb potentials. Additionally, they computed the spectrum for the Laplacian and its multiplicities when certain regions of a Laakso space are compressed or stretched and calculated the Casimir force experienced by two uncharged conducting plates by imposing physically relevant boundary conditions and then analytically regularizing the resulting zeta function. Lastly, they derived a general formula for the spectral zeta function and its derivative for Laakso spaces with strict self-similar structure before listing explicit spectral values for some special cases.
See also B. Steinhurst, Dirichlet Forms on Laakso and Barlow-Evans Fractals of Arbitrary Dimension arXiv:0811.1378

Publications

arXiv.org
AIP | Journal of Mathematical Physics

From Self-Similar Structures to Self-Similar Groups

April 14, 2016

Contributors

Daniel Kelleher, Benjamin Steinhurst, and Chuen-Ming Wong

Overview

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)

Modified Hanoi Towers Groups and Limit Spaces

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

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

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.

Publications

arXiv.org
Fractals