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We study chaotic systems generated by deterministic or probablistic mappings. We introduce the density function which is an eigenfunction of a probability-preserving kernel K. We are able to show that all eigenvalues of K have magnitude less than or equal to 1 and that the only magnitude-one eigenvalues are the Nth roots of unity. We have also calculated the corresponding eigenfunctions associated with these magnitude-one eigenvalues: These eigenfunctions can be expanded in terms of N positive functions having disjoint support. We then concentrate on a one-dimensional system, and study the behavior and mechanism for various chaotic transitions. We find that the mechanism associated with the 2 to 1 (or more generally, 2N to N) transition is different from those associated with other chaotic transitions. We then determine the conditions for these transitions, and express them in a universal form. We confirm the Huberman-Rudnick scaling in the large 2^n to 2^n-1 chaotic-transition region, and determine the prefactor at these transitions. In addition, we establish a simple relation between the Lyapunov exponent and the folding of the distribution functions. We have also studied the chaotic regions of this system numerically.
Chang et al. (Sun,) studied this question.