Abstract
In present article we consider a combinatorial problem of counting and classification of periodic orbits in dynamical systems on an example of the baker’s map. Periodic orbits of a chaotic system can be organized into a set of clusters, where orbits from a given cluster traverse approximately the same points of the phase space but in a different time-order.
We show that counting of cluster sizes in the baker’s map can be turned into a spectral problem for matrices from truncated unitary ensemble (TrUE). We formulate a conjecture of universality of the spectral edge in the eigenvalues distribution of TrUE and utilize it to derive asymptotics of the second moment of cluster distribution in the regime when both the orbit lengths and the parameter controlling closeness of the orbit actions tend to infinity. The result obtained allows to estimate the size of average cluster for various numbers of encounters in periodic orbit.
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Notes
In particular, if n is a prime number there are only two non-prime periodic orbits. They correspond to sequences of all zeroes or ones, respectively.
The rescaling of n by the factor 2n is equivalent to multiplication of each eigenvalue λ i (ϕ) by \(\sqrt{2}\). Note, that this rescaling only shifts the edge of the spectrum of matrices \({\mathcal{Q}}(\boldsymbol{\phi})\) to 1.
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Acknowledgements
We thank S. Kumar for valuable discussions and help with the derivation of Eqs. (4.3), (4.4). V.O. thanks Prof. T. Guhr for hospitality during his stay in Duisburg-Essen University. Financial support by the SFB/TR12 and Gu 1208/1-1 research grant of the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
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Gutkin, B., Osipov, V. Clustering of Periodic Orbits and Ensembles of Truncated Unitary Matrices. J Stat Phys 153, 1049–1064 (2013). https://doi.org/10.1007/s10955-013-0859-9
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DOI: https://doi.org/10.1007/s10955-013-0859-9