Consider a fixed point of an automorphism of a complex manifold of complex dimension 2. Such a fixed point is semi-parabolic if one of the multipliers is 1 and the modulus of the other multiplier is $\ne 1$. The local dynamical structure of a semi-parabolic, semi-attracting fixed point has been analyzed by Ueda and (later) by Hakim.

We consider the dynamics of perturbations of such maps which may be written in the form

$$f_\epsilon(x,y) = (x + x^2 -\epsilon^2 + \cdots, a y + \cdots)$$

If $|a|<1$ and $\epsilon=0$, then $(0,0)$ is a semi-parabolic and semi-attracting fixed point, and for small $\epsilon$, this fixed point splits into the pair $\sim(\pm\epsilon, 0)$. If $\epsilon\to0$ tangentially to the imaginary axis, then in an earlier paper with Smillie and Ueda, we found “implosion” phenomena, which explain certain discontinuities of the bifurcation.

Here we consider the situation where $\epsilon\to0$, but in a nontangential approach region, and we show that certain continuity phenomena hold. This is joint work in progress with Ueda.