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

Abstract

We study a certain class of classical one dimensional piecewise linear maps.\nFor these systems we introduce an infinite family of Markov partitions into\nequal cells. The symbolic dynamics generated by these systems is described by\nbistochastic (doubly stochastic) matrices. We analyze the structure of graphs\ngenerated from the corresponding symbolic dynamics. We demonstrate that the\nspectra of quantized graphs corresponding to the regular classical systems have\nlocally Poissonian statistics, while quantized graphs derived from classically\nchaotic systems display statistical properties characteristic of Circular\nUnitary Ensemble, even though the corresponding unitary matrices are sparse.\n