Book contents
- Frontmatter
- Contents
- Introduction
- Surveys
- Research Papers
- 1 Uniformity in the polynomial Szemérdi theorem
- 2 Some 2-d symbolic dynamical systems: Entropy and mixing
- 3 A note on certain rigid subshifts
- 4 Entropy of graphs, semigroups and groups
- 5 On representation of integers in Linear Numeration Systems
- 6 The structure of ergodic transformations conjugate to their inverses
- 7 Approximatiom by periodic transformations and diophantine approximation of the spectrum
- 8 Invariant σ-algebras for ℤd-actions and their applications
- 9 Large deviations for paths and configurations counting
- 10 A zeta function for ℤd-actions
- 11 The dynamical theory of tilings and Quasicrystallography
11 - The dynamical theory of tilings and Quasicrystallography
Published online by Cambridge University Press: 30 March 2010
- Frontmatter
- Contents
- Introduction
- Surveys
- Research Papers
- 1 Uniformity in the polynomial Szemérdi theorem
- 2 Some 2-d symbolic dynamical systems: Entropy and mixing
- 3 A note on certain rigid subshifts
- 4 Entropy of graphs, semigroups and groups
- 5 On representation of integers in Linear Numeration Systems
- 6 The structure of ergodic transformations conjugate to their inverses
- 7 Approximatiom by periodic transformations and diophantine approximation of the spectrum
- 8 Invariant σ-algebras for ℤd-actions and their applications
- 9 Large deviations for paths and configurations counting
- 10 A zeta function for ℤd-actions
- 11 The dynamical theory of tilings and Quasicrystallography
Summary
ABSTRACT. A tiling x of Rn is almost periodic if a copy of any patch in x occurs within a fixed distance from an arbitrary location in x. Periodic tilings are almost periodic, but aperiodic almost periodic tilings also exist; for example, the well known Penrose tilings have this property. This paper develops a generalized symmetry theory for almost periodic tilings which reduces in the periodic case to the classical theory of symmetry types. This approach to classification is based on a dynamical theory of tilings, which can be viewed as a continuous and multidimensional generalization of symbolic dynamics.
INTRODUCTION
The purpose of this this paper is to describe a natural generalization of the standard theory of symmetry types for periodic tilings to a larger class of tilings called almost periodic tilings. In particular, a tiling x of Rn is called almost periodic if a copy of any patch which occurs in x re-occurs within a bounded distance from an arbitrary location in x. Periodic tilings are clearly almost periodic since any patch occurs periodically, but there are also many aperiodic examples of almost periodic tilings—the most famous being the Penrose tilings, discovered in around 1974 by R. Penrose.
Ordinary symmetry theory is based on the notion of a symmetry group—the group of all rigid motions leaving an object invariant. The symmetry groups of periodic tilings are characterized by the fact that they contains a lattice of translations as a subgroup.
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- Ergodic Theory and Zd Actions , pp. 451 - 474Publisher: Cambridge University PressPrint publication year: 1996
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