next figure (Fig. \[FIG:repeller-onset\]), we chose the parameter near the emergence of the ring-like shaped repeller, and visualized the shape of the repeller and the attracting invariant set corresponding to the basin of repulsion of the repeller. Sequence of inward spikes are observed in the boundary curve of the basin. We think that these spikes are a symptom of a non-uniform saddle-node bifurcation, where the attractor of $\tilde M$ (which corresponds to the basin of the ring-like shaped repeller) collide with a saddle. Thus it is surmised that, at the bifurcation of the disappearance of the repeller, the boundary would touch the last component of the repeller on infinitely many fibers simultaneously, while on typical fibers the repeller of QPLM with fractal-like structure is still retained. Thus it is conjectured that the repeller would have strictly positive lyapunov exponent even at the onset.

![Repeller and its basin of repulsion, near the emergence of the repeller $(a,\epsilon)=(1.0,0.336057731)$. Data of 10000 target fibers are plotted together. \[FIG:repeller-onset\]](figure-3.eps){width="48.00000%"}

An interesting repetitive bifurcation structure has been reported near the torus doubling termination (TDT) critical point[@Kuznetsov-ETAL-98; @Kuznetsov-05]. Near this critical point, multiply nested structure of invariant sets is observed as shown in Figs.\[FIG:PD-TDT\] and \[FIG:multi-repellers\]. 