Beyond Quasicrystals: Les Houches, March 7–18, 1994 (Centre de Physique des Houches, 3)

This book is the collection of most of the written versions of the Courses given at the Winter School "Beyond Quasicrystals" in Les Houches (March 7-18, 1994). The School gathered lecturers and participants from all over the world and was prepared in the spirit of a general effort to promote theoretical and experimental interdisciplinary communication between mathematicians, theoretical and experimental physicists on the topic of the nature of geometric order in solids beyond standard periodicity and quasi periodicity. The overall structure of the book reflects the wish of the editors to pose this fundamental question of geometric order in solids from both the experimental and theoretical point of view. The first part is devoted more specifically to quasicrystals. These materials were the common starting point of most of the audience and present a first concrete example of a non-trivial geometric order. We chose to focus on a few fundamental aspects of quasicrystals related to hidden symmetries in solids which are not easily found in standard textbooks on the topic, not to reach an exhaustive survey which is already available elsewhere. Front Matter....Pages I-XVI Front Matter....Pages 1-1 Quasicrystals, Diophantine approximation and algebraic numbers....Pages 3-16 The pentacrystals....Pages 17-31 Elements of a multimetrical crystallography....Pages 33-54 Non-commutative Models for Quasicrystals....Pages 55-73 From Quasiperiodic to More Complex Systems....Pages 75-140 Matching Rules and Quasiperiodicity: the Octagonal Tilings....Pages 141-189 A mechanism for diffusion in quasicrystals....Pages 191-201 Experimental aspects of the structure analysis of aperiodic materials....Pages 203-228 Scattering on aperiodic superlattices....Pages 229-247 Defects in quasicrystals, in systems with deterministic disorder and in amorphous materials....Pages 249-290 Front Matter....Pages 291-291 Automata and automatic sequences....Pages 293-367 Spectral study of automatic and substitutive sequences....Pages 369-414 Random and automatic walks....Pages 415-432 Singular words, invertible substitutions and local isomorphisms....Pages 433-440 Entropy in deterministic and random systems....Pages 441-463 Trace maps....Pages 465-480 Schrödinger difference equation with deterministic ergodic potentials....Pages 481-549 Schrödinger equation in a hierarchical potential....Pages 551-561 Introduction to multifractal analysis....Pages 563-582 Back Matter....Pages 583-619 The topic of this book is the nature of geometric order in solids beyond standard periodicity and quasiperiodicity. Its overall structure reflects the wish of the editors to pose this fundamental question from both the experimental and theoretical point of view. The first part is book is devoted more specifically to quasicrystals. The second part deals with deterministic aperiodic order "beyond" quasiperiodicity, as generated by automatic and substitutive sequences. This pedagogically written topical volume can be seen by experimentalists as a manual on the theoretical background of their work This work presents geometric and topological properties of quasicrystals and studies diffraction theory of long-range objects with non-trivial order. The book aims to provide a rigorous framework for sequences generation and Fourier analysis.