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Spectral decomposition of tent maps using symmetry considerations. (English) Zbl 1081.37501
Summary: The spectral decomposition of the Frobenius-Perron operator of maps composed of many tents is determined from symmetry considerations. The eigenstates involve Euler as well as Bernoulli polynomials.
MSC:
37A30Ergodic theorems, spectral theory, Markov operators
37C25Fixed points, periodic points, fixed-point index theory
37C30Functional analytic techniques in dynamical systems
References:
[1]A. Lasota and M. Mackey,Probabilistic properties of Deterministic Systems (Cambridge University Press, Cambridge, 1985).
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[3]H. H. Hasegawa and W. C. Saphir, Unitarity and irreversibility in chaotic systems,Phys. Ret. A 46:7401 (1992); P. Gaspard,r-Adic one-dimensional maps and the Euler summation formula,J. Phys. A 25:L483 (1992). · doi:10.1103/PhysRevA.46.7401
[4]H. H. Hasegawa and D. J. Driebe, Intrinsic irreversibility and the validity of the kinetic description of chaotic systems.Phys. Rev. E 50:1781 (1994); I. Antoniou and S. Tasaki, Generalized spectral decomposition of the β-adic baker’s transformation and intrinsic irreversibility,Physica A 190:303 (1992): I. Antoniou and S. Tasaki, Generalized spectral decompositions of mixing dynamical systems,Int. J. Quantum Chem.46:425 (1993). · doi:10.1103/PhysRevE.50.1781
[5]D. J. Driebe and G. E. Ordóñez, Using symmetries of the Frobenius-Perron operator to determine spectral decompositions,Phys. Lett. A 211:204 (1996). · Zbl 1060.37500 · doi:10.1016/0375-9601(96)00006-0
[6]P. A. M. Dirac,The Principles of Quantum Mechanics (Oxford University Press, London, 1958).
[7]M. Abramowitz and I. A. Stegun, eds.Handbook of Mathematical Functions (Dover, Publications New York, 1972).