# Dynamics seminar

Title: Some new minimal homeomorphisms, why they shouldn't exist and why it's important that they do.

Speaker: Ian Putnam, University of Victoria

Date and time:
26 Feb 2015,
2:30pm -
3:30pm

Location: Clearihue A127

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It is an old question in dynamical systems: which compact metric spaces admit minimal (uniquely ergodic) homeomorphisms? The circle obviously has irrational rotations, but for higher dimensional spheres, this question becomes quite interesting. One of the major positive results is by Fathi and Herman which asserts that odd dimensional spheres admit minimal, uniquely ergodic homeomorphisms. On the other hand, the most famous negative result is the Hopf-Lefschetz theorem which, in particular, shows that even dimensional spheres do not. After discussing these, I will give an example of another space which admits a minimal, uniquely ergodic homeomorphism, managing in a crafty manner to avoid the Hopf-Lefschetz theorem, which says that it shouldn't. This has interesting consequences for George Elliott's classification program for C*-algebras. I'll discuss these, without assuming any prior knowledge of C*-algebras.

Title: Gibbs measures, hard constraints and (boundary) phase transitions

Speaker: Raimundo Briceño, UBC

Date and time:
18 Feb 2015,
2:30pm -
3:30pm

Location: Cornett A128

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Inspired by topological notions coming from symbolic dynamics, we introduce a new combinatorial condition (TSSM) on the support of Z^d Markov random fields, especially useful when dealing with hard constrained systems. We establish some of its properties and different characterizations. We also show how TSSM is related with the absence of phase transitions in the context of nearest-neighbour Gibbs measures.

Title: Block Triangularization of Matrix Cocycles

Speaker: Joseph Horan, University of Victoria

Date and time:
05 Feb 2015,
1:30pm -
2:30pm

Location: DSB C126

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When dealing with matrices, putting them into a triangular form often yields meaningful information about their structure. In the case of invertible real matrix cocycles over an invertible, ergodic map, the Multiplicative Ergodic Theorem provides a subspace decomposition of $\mathbb{R}^n$, which may be interpreted as a statement about block diagonalizing the cocycle. One can then ask if cocycles may always be block triangularized, not just block diagonalized; we answer this question in the negative, and therefore show that the MET is in this sense optimal.

Title: A Minimal Subsystem of the Kari Culik Tilings

Speaker: Jason Siefken, University of Victoria

Date and time:
29 Jan 2015,
1:30pm -
2:30pm

Location: DSB C126

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The set of 13 Kari Culik tiles is currently the smallest know set of Wang tiles (square tiles with colored edges and the rule that two tiles may lie adjacent iff their common edges share the same color) that tile the plane only in an aperiodic way. They do so for fundamentally different reasons than previously known tilings. A subset of Kari Culik tilings have rows which may be interpreted as Sturmian sequences. This talk will show how this Sturmian-like subset can be thought of as a generalization of rotation sequences and how to get explicit waiting time bounds for n*m configurations.