and check if the length of the segment would converge to 0 or not. If it converges to 0, that indicates no transversely unstable point is contained in the initial segment. On the other hand, if the length of the orbit starting from the segment has a positive limsup value, transversely unstable point(s) presumably exist in the initial segment.

The asymptotic behavior of the orbit of the segment map gives some information about the recurrence property of the points in the initial segment as well. As the $\omega$-limit set of an orbit of the segment map is an invariant set, its shadow determines a corresponding invariant set of the original QPLM. If the initial segment (i.e., a segment in QPLM corresponding to the shadow of the initial point of the segment map) is a subset of this invariant set, there should exist non-wandering points in the initial segment.

Thus we are interested in such point ($p$) that satisfies

-   $\omega$-limit set of any orbits (of $\tilde M$) starting from a point corresponding to non-zero length segment containing the point $p$ is detached from $w=0$ plane,

-   the shadow of the $\omega$-limit set contains the point $p$

Above mentioned $\omega$-limit set 