and for the segment map system). For a point $u=(\theta,z,w)\in\tilde\Omega$, the length of the corresponding segment in $\Omega$ ($= 2w$) is written as ${{\rm length}\ }(u)$. For a point $u=(\theta,z,w)$ in $\tilde \Omega$, we let $\underline u$ to represent a set (segment) $\{(\theta,x)\in\Omega
\ |\ z-w\leq x\leq z+w\}$. Similarly, for $U$ which is a subset of $\tilde\Omega$, let $\underline U$ stands for $\bigcup_{u\in
  U}\underline u \subset \Omega$. We call $\underline U$ as the “shadow” of $U$. For an invariant set (that may or may not be an attractor) $X\in W$ where $W$ may be either $\Omega$ or $\tilde\Omega$, we let $B(X)$ denote the basin of attraction $\{u\in W
\ |\ \omega(u)\subset X\}$, where $\omega(u)$ denotes the $\omega$-limit set of the orbit starting from $u$.

A method to locate repellers
----------------------------

Now let us describe the outline of the method to detect and to locate “repellers”. Our basic strategy is to find a set of points that are transversely unstable and non-wandering, using the map describing temporal evolution of a line segment.

As for the transverse stability of a point, we observe asymptotic behavior of an orbit (of the segment map) starting from a segment (with positive length) containing the point, 