Basic dynamical stability results for

Abstract

AbstractWe study stability questions for dynamics on $\mathbb {P}^n$ using holomorphic families, which we require to be locally simply parameterized. Given such a holomorphic family $f\colon \mathbb {P}^n\times M\to \mathbb {P}^n$, we define the notion of a ‘Fatou section’, which is intuitively a choice of one point in $\mathbb {P}^n$ for each map in the family such that: (1) the chosen point depends holomorphically on M, and (2) the points chosen for two different maps behave dynamically ‘comparably’ under iteration of those respective maps. We prove a weak version of the λ-lemma for dynamical systems on $\mathbb {P}^n$ by showing that the set of all Fatou sections is a compact space. We introduce the notion of a postcritically bounded holomorphic family and show that if a (locally simply parameterized) holomorphic family is postcritically bounded by some open subset of $\mathbb {P}^n$ then: (1) repelling periodic points in the Julia set for one member of the family cannot bifurcate, nor can they leave the Julia set in other members of the family, (2) there is a Fatou section which never leaves the Julia set through each point of the Julia set of any member of the family, and (3) any intersection between the Julia set and the critical set is persistent, in the sense that there is a Fatou section through the point of intersection which never leaves either the critical set or the Julia set. Here, for conciseness, we are using the term ‘Julia set’ to mean the support of the unique measure of maximal entropy, although we recognize that how this term should be used in higher dimensions is still undecided. We also provide a brief summary of the development of the field in our introduction.