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 $f^l\mathrm{}\left(H\right),$ 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 $\epsilon .$