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Chaotic backward shift operator on Chebyshev polynomials

Published online by Cambridge University Press:  21 December 2018

MÁRTON KISS
Affiliation:
Department of Differential Equations, Institute of Mathematics, Budapest University of Technology and Economics, H 1111 Budapest, Müegyetem rkp. 3-9, Hungary email: [email protected]
TAMÁS KALMÁR-NAGY
Affiliation:
Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics, H 1111 Budapest, Müegyetem rkp. 3-9, Hungary email: [email protected]

Abstract

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

Type
Papers
Copyright
© Cambridge University Press 2018 

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