J=IntialTransient(I)
      if (J is not in InvList)
        then
          append J in InvList
          if length(J)>0
            then 
              Unity[J]=P1UnityCheck(I,J)
            else 
              Unity[J]=true
      if ( (Unity[J] is false)
          and (length(I) > param_cutoff) )
        then 
          divide I into (I_1, I_2)
          call P1CheckImage(I_1)
          call P1CheckImage(I_2)

    boolean P1UnityCheck(I,J) 
      "Take an arbitrary point v in segment I"
      if ("LyapunovExponent(v)"<0)
        then 
          return false
      "Take a segment V, as
         theta(V)=theta(J),
         center(V)=InitialTransient(v),
         length(V)=very short"
      if (IterateMany(V) and
          IterateMany(J) coincide)
        then
          return true
        else
          return false

In this part, we try to obtain a list of pull back attractors of $\tilde M$ on a target fiber. We are interested in such attractors which are also attractors of $M$ (i.e., stable torus/tori, SNA, chaotic attractor) or which attract orbits with initial conditions that correspond to short segments representing sufficiently small neighborhoods of a point on repellers of $M$.

Let the target fiber be specified as $\{\theta=\theta_0\}$ ($\theta_0\in[0,1)$), $C_-$ be a constant satisfying $a-C_-^2
+\epsilon < C_-$, and $\tau^-$ be a sufficiently large integer parameter that gives the duration of initial transient. Then we take “initial segment” (a point in $\tilde\Omega$) $I_{init}=(\theta_-,0,|C_-|)$ on the initial fiber specified with $\theta_-=(\theta_0 - \tau^- \omega)\ \rm{mod}\ 1$. With this choice of $I_{init}$, every invariant set of $M$ should 