Dynamics of polynomials with disconnected Julia sets (Q1395980)

The author gives a sufficient combinatorial condition for the Julia set of a nonhyperbolic polynomial to be a zero-area Cantor set. In fact, the main result of this paper is that given natural numbers \(m_1,\ldots,m_k\), there are uncountably many combinatorially inequivalent polynomials \(p\) such that the Julia set of \(p\) is a zero-area Cantor set and \(p\) has \(k\) nonescaping critical points \(\omega_1,\ldots,\omega_k\), with respective multiplicities \(m_1,\ldots,m_k\), such that \(\omega_i\) accumulates at \(\omega_j\) for all \(1\leq i,j \leq k\). The engine of the proof is the following theorem [\textit{C. T. McMullen}, Complex dynamics and renormalization, Annals of Mathematics Studies, 135, Princeton,NJ: Univ. Press. (1995; Zbl 0822.30002)]: Let \(U_n\subset \mathbb{C}\) be a sequence of disjoint open sets such that each \(U_n\) is a finite union of disjoint unnested annuli of finite modulus, each component of \(U_{n+1}\) is nested inside a component of \(U_n\) and \(\sum \bmod A_n=\infty\) for any sequence of nested annuli \(A_n\subset U_n\). If \(B_n\) denotes the bounded components of \(\mathbb{C}\setminus U_n\), then \(\bigcap B_n\) is a Cantor set of area zero. The author defines an infinite modulus condition for trees with dynamics, which were introduced in [\textit{R. Pérez-Marco}, Degenerate conformal structures, Manuscript (1999)]. In case such an abstract tree is realized by a polynomial, then its Julia set is contained in the complement of the union of nested sets of disjoint annuli. Thus, by McMullen's theorem, the Julia set is a zero-area Cantor set. Finally, the author shows that every tree with dynamics is realized by a polynomial if it allows certain invariants compatible with the dynamics. Especially, given natural numbers \(m_1,\ldots,m_k\), one can construct a tree with dynamics in such a way that it has the infinite modulus condition, is realized by a polynomial and this polynomial exhibits the properties stated in the main theorem.