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

Abstract:

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.