Quasiperiodicity and types of order; a study in one dimension

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Abstract
In order to characterise the interplay between quasiperiodicity and order in one dimension, the authors consider sequences of 0 and 1 generated by a circle map. These sequences, which generalise the Fibonacci sequence, describe the quasiperiodic ordering of atoms and vacancies on a line. They study in a quantitative way the unbounded fluctuation of the atomic positions WRT the average lattice. For some quadratic algebraic values of the rotation number the sequences can be generated by inflation rules, which proves their self-similarity. These rules, obtained by a renormalisation of the circle map generating the sequences, permit us, e.g. to explain the logarithmic divergence of the fluctuation. For some exceptional rotation numbers, the fluctuation diverges as N alpha , N being the system size. Whenever alpha >1/2, the quasiperiodic chain is therefore less 'rigid' than a random one.