have non-empty intersection with $\underline{I_{init}}$.

We calculate the trajectory starting from a segment $I$ (which is set as $I_{init}$ for the first trial) on the initial fiber to obtain its image ($J$) on the target fiber, then record $J$ as a member of the list of pull back attractor on the target fiber (“InvList”). If it has non-zero length and does not correspond to “transitive” chaotic attractor, it is presumed that there exist pull back attractor(s) which corresponds to a proper subset of $J$. In such case, we divide the initial segment $I$ into two, and recursively repeat this calculation with using each fragment of $I$ as initial segment, until the initial segment becomes sufficiently short.

#### Part 2: Visualization of repellers

    Part2Main
      foreach J in InvList
        if (Unity[J] is false)
          then 
            foreach t in TList
              clear tmpRepellerImage
              JT="Iterate_t_times(J)" 
              call P2Decomposition(JT)
              RepellerImage[J,t]=tmpRepellerImage


    Procedure P2Decomposition(J)
      K=IterateManytimes(J)
      call P2DecompSub(J,K)

    Procedure P2DecompSub(JX,K)
      L=IterateManytimes(JX)
      if (L matches K)
        then
          if (length(JX)<param_cutoff)
          then
            "append JX to tmpRepellerImage"
          else
            "divide JX into (JX1,JX2)"
            call P2DecompSub(JX1,K)
            call P2DecompSub(JX2,K)

In this part, we choose multiple target fibers, and try to obtain the image of each repeller on each target fiber. The target fibers are chosen to be located 