Transitions and distribution functions for chaotic systems

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 $N\mathrm{th}$ 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 $2^{n-1\mathrm{}}$ 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.