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Periodic Orbit Action Correlations in the Baker Map Gregor Tanner Basic Research Institute in the Mathematical Sciences HP Laboratories Bristol HPL-BRIMS-1999-03 April, 1999 E-mail: [email protected] random matrix theory, periodic orbit correlation, quantum chaos Periodic orbit action correlations are studied for the piecewise linear, area-preserving Baker map. Semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum Baker map suggest the existence of universal periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius type operator. This makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map. Periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error. Exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation.

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Page 1: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius

Periodic Orbit Action Correlationsin the Baker Map

Gregor TannerBasic Research Institute in the Mathematical SciencesHP Laboratories BristolHPL-BRIMS-1999-03April, 1999

E-mail: [email protected]

random matrixtheory,periodic orbitcorrelation,quantum chaos

Periodic orbit action correlations are studied forthe piecewise linear, area-preserving Baker map.Semiclassical periodic orbit formulae together withuniversal spectral statistics in the correspondingquantum Baker map suggest the existence of universalperiodic orbit correlations. The calculation of periodicorbit sums for the Baker map can be performed withthe help of a Perron-Frobenius type operator. Thismakes it possible to study periodic orbit correlationsfor orbits with period up to 500 iterations of themap. Periodic orbit correlations are found to agreequantitatively with the predictions from randommatrix theory up to a critical length determined by thesemiclassical error. Exponentially increasing termsdominate the correlations for longer orbits which aredue to the violation of unitarity in the semiclassicalapproximation.

Page 2: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 3: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 4: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 5: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 6: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 7: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 8: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 9: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 10: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 11: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 12: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 13: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 14: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius
Page 15: Periodic Orbit Action Correlations in the Baker Map · periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius

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