Classical 1D maps, quantum graphs and ensembles of unitary matrices

Details Classical 1D maps, quantum graphs and ensembles of unitary matrices Classical 1D maps, quantum graphs and ensembles of unitary matrices

2000-11-30

arxiv.org/abs/nlin/0011050v5

J. Phys. A: Math. Gen. 34 (2001) 9303-9317

Nonlinear Sciences

Chaotic Dynamics

REVTeX 13 pages, 12 figures included, final published version

Scientific paper

10.1088/0305-4470/34/43/313

We study a certain class of classical one dimensional piecewise linear maps. For these systems we introduce an infinite family of Markov partitions into equal cells. The symbolic dynamics generated by these systems is described by bistochastic (doubly stochastic) matrices. We analyze the structure of graphs generated from the corresponding symbolic dynamics. We demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally Poissonian statistics, while quantized graphs derived from classically chaotic systems display statistical properties characteristic of Circular Unitary Ensemble, even though the corresponding unitary matrices are sparse.

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