Pentagonal domain exchange
Shigeki Akiyama Edmund Harriss
Discrete & Continuous Dynamical Systems - A 2013, 33(10): 4375-4400 doi: 10.3934/dcds.2013.33.4375
Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section.
keywords: numeration systems. pisot number piecewise isometry symbolic dynamics Pentagon
Strong coincidence and overlap coincidence
Shigeki Akiyama
Discrete & Continuous Dynamical Systems - A 2016, 36(10): 5223-5230 doi: 10.3934/dcds.2016027
We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for dimension $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper [2] and elucidates the tight relationship between two coincidences.
keywords: substitution Pisot number. pure discrete spectrum Self-affine tiling coincidence

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