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Baker's Map

The map

x_(n+1)=2mux_n,
(1)

where x is computed modulo 1. A generalized Baker's map can be defined as

x_(n+1)={lambda_ax_n y_n<alpha ; (1-lambda_b)+lambda_bx_n y_n>alpha
(2)
y_(n+1)={(y_n)/alpha y_n<alpha ; (y_n-alpha)/beta y_n>alpha,
(3)

where beta=1-alpha, lambda_a+lambda_b<=1, and x and y are computed mod 1. The q=1 q-dimension is

D_1=1+(alphaln(1/alpha)+betaln(1/beta))/(alphaln(1/(lambda_a))+betaln(1/(lambda_b))).
(4)

If lambda_a=lambda_b, then the general q-dimension is

D_q=1+1/(q-1)(ln(alpha^q+beta^q))/(lnlambda_a).
(5)

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References

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 60, 1983.Ott, E. Chaos in Dynamical Systems. Cambridge, England: Cambridge University Press, pp. 81-82, 1993.Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 32, 1990.

Referenced on Wolfram|Alpha

Baker's Map

Cite this as:

Weisstein, Eric W. "Baker's Map." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BakersMap.html

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