Naturally invariant measure of chaotic attractors and the conditionally invariant measure of embedded chaotic repellers.
We study local and global correlations between the naturally invariant measure of a chaotic one-dimensional map f and the conditionally invariant measure of the transiently chaotic map f(H). The two maps differ only within a narrow interval H, while the two measures significantly differ within the images f(l)(H), where l is smaller than some critical number l(c). We point out two different types of correlations. Typically, the critical number l(c) is small. The chi(2) value, which characterizes the global discrepancy between the two measures, typically obeys a power-law dependence on the width epsilon of the interval H, with the exponent identical to the information dimension. If H is centered on an image of the critical point, then l(c) increases indefinitely with the decrease of epsilon, and the chi(2) value obeys a modulated power-law dependence on epsilon.
Buljan
Hrvoje
H
Department of Physics, Faculty of Science, University of Zagreb, PP 332, 10000 Zagreb, Croatia.
Paar
Vladimir
V