Fiber Julia sets of polynomial skew products with super-saddle fixed points

If a polynomial skew product on \mathbb{C}^2 has a relation between two saddle fixed points, fiber Julia sets J_z behave discontinuously. That is, as the base variable z tends to a point \beta corresponding to a saddle point, the limits of J_z strictly include J_{\beta }. When the map is linearizable at these saddle points, we have described their behaviors in terms of Lavaurs maps in [Indiana Univ. Math. J. 68 (2019), pp. 35-61]. In this article, we consider the case when the map is not invertible at a saddle fixed point. It turns out that the Lavaurs map must be identically zero. As a result, the limits of fiber Julia sets have non-empty interiors.