Date | Speaker | Title |
---|---|---|
14/1/2011 | Jason Siefken (UVic) | Ergodic Optimization of super-continuous functions |
21/1/2011 | Michael Schraudner (Univ of Chile) | Projective subdynamics of Z^d shifts of finite type |
28/1/2011 | Antoine Julien (UVic) | Complexity for cut and project tilings |
4/2/2011 |   |   |
11/2/2011 |   | No seminar |
17/2/2011 | Ronnie Pavlov (Denver University) | Shifts of finite type with nearly full entropy |
25/2/2011 |   | No seminar |
4/3/2011 | Robert Moody (UVic) | On Joan Taylor's hexagonal monotile |
11/3/2011 | Tom Meyerovitch (UBC) | Ergodicity of Poisson Products |
18/3/2011 | Mike Hochman (Microsoft) | Geometric rigidity of times-m invariant measures |
25/3/2011 | Ayse Sahin (DePaul) | The directional entropy of ergodic, zero entropy Z^d actions and the loosely Bernoulli property |
31/3/2011 at 1:20 in C130 | John Griesmer (UBC) | Dense sets of integers via dynamics |
In this talk, I will present the cut and project method, which was used to produce many interesting example, like the Penrose tiling or the Octogonal tiling. The Penrose tiling itself is a model for some quasi-crystals. Tilings produced by this method, while non periodic, are very much "ordered", and are very far from being random tilings.
We measure this "aperiodic order" by using the complexity function, which counts the number of different patterns of a given size. The theorem that I want to present gives the asymptotic order of growth of the complexity function. This result can also be seen as a multi-dimensional generalization of the computation of complexity for cubic billiard sequences.
Z^d-SFTs are generally much more complicated than their one-dimensional counterparts; one illustration of this is given by their measures of maximal entropy. Any topologically mixing Z-SFT has a unique m.m.e., but it was shown by Burton and Steif that not even uniform topological mixing conditions such as strong irreducibility (which often suffice to generalize theorems about Z-SFTs to Z^d) imply that a Z^d-SFT has a unique m.m.e.
We present a new sufficient condition for a Z^d-SFT to have a unique m.m.e., which is expressed purely in terms of topological entropy. Namely, for any d, there is a constant B(d) > 0 so that any nearest-neighbor Z^d-SFT X with alphabet A and entropy at least (log |A|) - B(d) has a unique m.m.e. We will also present some examples and background to illustrate how this result compares to other sufficient conditions in the literature.
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