non-trivial structure. Summary and short discussion will be given in the last section.

Method
======

In this section, we will explain our method to obtain the images of “repellers” in QPLM. In the first place, we introduce concrete form of the system and auxiliary “segment map”, a 1+2- dimensional skew-product map that describes the evolution of a “segment” in the phase space of QPLM. Then we describe the method to obtain repellers in QPLM. Some technical details of the algorithm to obtain the images of repellers, together with a method to distinguish SNA from “smooth” torus will be given in appendix.

QPLM and “segment map”
----------------------

Here we consider the quasi-periodically forced logistic (quadratic) map in the following form, $M|\Omega\rightarrow\Omega$,$(\Omega:=T^1([0,1))\times{\mathbf R})$, $M(\theta_n,x_n)=(\theta_{n+1},x_{n+1})$ $$\begin{aligned}
\theta_{n+1} &=& (\theta_n + \omega) \ \mbox{mod}\ 1,\cr
x_{n+1} &=& a - x_n^2 + \epsilon \cos(2\pi\theta_n).
\label{EQ:QPLM}\end{aligned}$$ $a$ and $\epsilon$ are treated as control parameters, and $\omega$ is fixed as $(\sqrt{5}-1)/2$ in this paper.

In order to examine the stability of a point in $\Omega$, it would be a natural approach to consider the temporal evolution of its neighborhood. Regarding the trivial neutral stability in the $\theta$ direction, it seems reasonable to think about the evolution of 