stable and unstable manifold intersect, then $\Phi(x) = 0$ is allowed and the strange attractors are associated to homoclinic intersections. For the modern theory of chaos and dynamical systems, this is a case of historical and practical importance; see [@GH; @SSTC1; @SSTC2]. To this case, unfortunately, the theory of rank one maps in [@WY3] does not apply because of the existence of the singularities of $f_{a}$. Our ultimate goal is to develop a theory that can be applied to the separatrix maps allowing $\Phi(x) = 0$. This paper is the first step, in which we develop a 1D theory.

For $f = f_{a}$, let $C(f) = \{ f'(x) = 0 \}$ be the set of critical points and $S(f) = \{ \Phi(x) = 0 \}$ be the set of singular points. In this paper we are interested in the case $L\gg1$. As $L$ gets larger, the contracting region gets smaller and the dynamics is more and more expanding in most of the phase space. Nevertheless, the recurrence of the critical points is inevitable, and thus infinitesimal changes of dynamics occur when $a$ is varied. In addition, the logarithmic nature of the singular set $S$ turns out to present a new 