appreciate N. Takahashi, T. Mitsui, and Y. Sato for helpful suggestion and discussion. This research is supported by JSPS KAKENHI Grant Number 60115938.

numerical algorithm
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Here we describe some details of the algorithm used in our numerical calculation. Our calculation consists of 3 parts. (1) Search for the invariant sets with nested structure, which gives information of the existence/absence of repellers. (2) Decomposition of the invariant sets into the repeller and the basin of smaller attracting invariant sets, to obtain the approximate images of the repellers. (3) Check for the behavior of perturbed orbits starting from a point on the attractors obtained in (1), in order to distinguish “smooth” attracting torus and SNA. Both (2) and (3) need the result of (1), but (2) and (3) could be carried out mutually independently.

In part (1), we fix a target fiber and will make a list of pull back attractors of $\tilde M$ on the fiber. Here we intend to list up all the attractors that may attract some of the sufficiently short vertical segments. The obtained list would contain those corresponding to the basin of repulsion of repellers as well as those directly corresponding to the attractors of $M$. 