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Shadowing and iterative interpolation for Čebyšev mixing transformations. (English) Zbl 0756.58037

Summary: The iteration of Chebyshev polynomials generates mixing transformations that model canonical features of chaotic systems. These include pseudo- random evolution, ergodicity, fading memory, and the irreversible dispersal of any set of positive measure throughout the mixing region. Mixing processes are also analytically and numerically unstable. Nevertheless, iterative interpolation, or numerical retrodiction, demonstrates that the computer generated trajectories are shadowed within strict error bounds by exact Chebyshev iterates. Pervasive shadowing is, however, not sufficient to ensure a generic correspondence between computer simulations and “true dynamics”. This latitude is illustrated by several basic distinctions between the computer generated orbit structures and the exact analytic orbits of the Chebyshev mixing transformations.

MSC:

37D99 Dynamical systems with hyperbolic behavior
41A50 Best approximation, Chebyshev systems
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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