infinitely many apertures (with progressively thin widths), and thus when adequately coarse grained it could appear to consist of finite number of chunks. Thus the approximation for the repeller obtained by RA could be a good approximation for the repeller in quasi-periodically forced system, in spite of the fact that the set of finite number of rings cannot be invariant in the quasi-periodically forced system.

![An example of repeller with non-trivial structure observed at $(a,\epsilon)=(1.0,0.35)$. Thin apertures in the rim of ring like structures are visible in enlarged view in boxes (b) and (c). Data of pull-back attractors and repellers on 10000 target fibers in $\theta\in[0,1]$ are used. \[FIG:typical-example\]](figure-2a-s.eps "fig:"){width="48.00000%"}![An example of repeller with non-trivial structure observed at $(a,\epsilon)=(1.0,0.35)$. Thin apertures in the rim of ring like structures are visible in enlarged view in boxes (b) and (c). Data of pull-back attractors and repellers on 10000 target fibers in $\theta\in[0,1]$ are used. \[FIG:typical-example\]](figure-2b-s.eps "fig:"){width="48.00000%"}![An example of repeller with non-trivial structure observed at $(a,\epsilon)=(1.0,0.35)$. Thin apertures in the rim of ring like structures are visible in enlarged view in boxes (b) and (c). Data of pull-back attractors and repellers on 10000 target fibers in $\theta\in[0,1]$ are used. \[FIG:typical-example\]](figure-2c-s.eps "fig:"){width="48.00000%"}

In the 