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arXiv: 1012.1222

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Journées Automates Cellulaires 2010, Turku : Finland (2010)

Computing (or not) Quasi-periodicity Functions of Tilings

Alexis Ballier 1, Emmanuel Jeandel 1

(2010-12-15)

We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic tiling. We prove that the tilings by a tileset that admits only quasi-periodic tilings have a recursively (and uniformly) bounded quasi-periodicity function. This corrects an error from [6, theorem 9] which stated the contrary. Instead we construct a tileset for which any quasi-periodic tiling has a quasi-periodicity function that cannot be recursively bounded. We provide such a construction for 1-dimensional effective subshifts and obtain as a corollary the result for tilings of the plane via recent links between these objects [1, 10].

1:	Laboratoire d'informatique Fondamentale de Marseille (LIF)
	CNRS : UMR6166 – Université de la Méditerranée - Aix-Marseille II – Université de Provence - Aix-Marseille I

Subject

:

Computer Science/Other

Keyword(s): tilings – multidimensional symbolic dynamics – quasi-periodicity

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From: Pierre Guillon <>

Submitted on: Thursday, 2 December 2010 17:16:17

Updated on: Monday, 6 December 2010 12:43:47