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

We study a certain class of classical one-dimensional piecewise linear maps. For these systems we introduce an infinite family of Markov partitions in equal cells. The symbolic dynamics generated by these systems is described by bi-stochastic (doubly stochastic) matrices. We analyse 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 the circular unitary ensemble, even though the corresponding unitary matrices are sparse.