Baker's Map

The Map

$$x_{n+1} = 2\mu x_n,$$ (1)

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

$$x_{n+1} = \left\{\begin{array}{ll}\lambda_ax_n & \mbox{$y_n< \alpha$\ }\\ (1-\lambda_b)+\lambda_bx_n & \mbox{$y_n>\alpha$\ }\end{array}\right.$$ (2)

$$y_{n+1} = \left\{\begin{array}{ll}{y_n\over \alpha} & \mbox{$y_n<\alpha$\ }\\ {y_n-\alpha \over \beta } & \mbox{$y_n>\alpha,$}\end{array}\right.$$ (3)

where $\beta \equiv 1-\alpha$, $\lambda_a+\lambda_b\leq 1$, and $x$ and $y$ are computed mod 1. The $q=1$ q-Dimension is

$$D_1 = 1+{\alpha\ln\left({1\over\alpha}\right)+\beta\ln\left(... ...ver\lambda_a}\right)+\beta \ln\left({1\over\lambda_b}\right)}.$$ (4)

If $\lambda_a=\lambda_b$, then the general q-Dimension is

$$D_q = 1+{1\over q-1} {\ln\left({\alpha^q+\beta^q}\right)\over\ln\lambda_a}.$$ (5)

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.