For a two-dimensional piecewise linear map with a riddled basin, a multifractal spectrum which characterizes the “skeletons” of the riddled basin, is introduced. With the uncertainty exponent is obtained by a variational principle, which enables us to introduce a concept of a “boundary” for the riddled basin. A conjecture on the relation between and the “stable sets” of various ergodic measures, which coexist with the natural invariant measure on the chaotic attractor, is also proposed.
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See, for example, C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems. An Introduction (Cambridge University Press, Cambridge, 1993).
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See, for example, P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998);
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18.
A proof of the large deviation law with defined by the variational principle Eq. (20) for hyperbolic systems can be found in Ref. 16.
19.
20.
21.
For a system of the form as Eq. (10), at assume that the D-Lebesgue measure of on a line obeys for small l and take an ergodic measure with and Consider a horizontal line segment of size across the stable set of the ergodic measure at and take the forward image of this line segment m times such that For this image, the distance from is For sufficiently small and l, the 1D-Lebesgue measure of on the line segment is considered to be approximated by which leads to Eq. (22).
22.
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993).
23.
If the basin is riddled, it is no longer an open set and its boundary is not well-defined.
24.
25.
Note that, in nonhyperbolic systems, the equivalence between the spectrum of fluctuation and the escape rate from the ergodic measure does not always hold obviously for the fluctuation with the negative value of while it is expected to hold partly for the hyperbolic part of the fluctuation (Ref. 17).
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