(Fig. \[FIG:fractalized\]), the parameter is set as $(a,\epsilon)=(0.75,0.46672)$, a little after the onset of SNA by “fractalization” on the path corresponding to the one investigated by Kaneko and Nishikawa[@Kaneko-84b; @Nishikawa-Kaneko-96]. We can see a kind of fractal like complicated structure with some apparent smoothness, however, the images of the repeller is “dirty” due to apparently superimposed irregular vertical stripe like structure. We think that the repeller would be dense in its basin of repulsion due to these stripes and correspondingly the basin of SNA in $\tilde M$ should be riddled, while structures with some smoothness would be apparent as far as we observe finite number of fibers.

![Repeller and SNA emerged via fractalization route, $(a,\epsilon)=(0.75, 0.46673)$. The pull-back attractor (SNA) and the repeller, as well as boundary of absorbing areas, are plotted for 4000 (a), 2000 (b and c) target fibers, with $\theta$ values in $[0,1]$ for (a), $[0.8,0.9]$ for (b), and $[0.84,0.85]$ for (c) respectively.\[FIG:fractalized\]](figure-6a.eps "fig:"){width="46.00000%"} ![Repeller and SNA emerged via fractalization route, $(a,\epsilon)=(0.75, 0.46673)$. The pull-back attractor (SNA) and the repeller, as well as boundary of absorbing areas, are plotted for 4000 (a), 2000 (b and c) target fibers, with $\theta$ values in $[0,1]$ for (a), $[0.8,0.9]$ for 