Some rational maps whose Julia sets are not locally connected

Abstract

Abstract. We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected.

A brief introduction to one-dimensional complex dynamics

“…Douady 和 Hubbard 在 20 世纪 80 年代构造了一个无穷 satellite 可重整且 Julia 集非局部连通的二次多项式 [516] . 更多非局部连通 Julia 集的例子 19) 可参见 文献 [249,434,644,702].…”

“…目前所有已知的连通且非局部连通的有理函数 Julia 集, 造成非局部连通的原因都是某个 Fatou 分支的边界非局部连通, 而不是因为 Fatou 分支的直径不趋向于 0 (参见引理 2.6). Roesch [644] 对于没有无理中性周期点且 Julia 集连通的多项式, 其 Julia 集局部连通当且仅当 Julia 集上没有 游荡连续统 (continuum) 20) (参见文献 [412, 第 245 页] 和 [433,729]).…”

Buried points in Julia sets

“…However, there are connected polynomial Julia sets which are not locally connected [Mil06,Corollary 18.6]. Recently, Pascale Roesch showed that there are "genuine" rational Julia sets that are connected but not locally connected [Roe06]. The rational map is not conjugate to a polynomial, but the non-local-connectivity is achieved through reference to polynomials.…”

Two-dimensional Blaschke products: degree growth and ergodic consequences

“…Note that if one allows some of the zeros e i of a one-variable Blaschke product (1) to have modulus greater than 1, a much more complicated structure for the Julia set can occur [24]. We do not consider the 2-variable analog of that situation in this paper, but we expect that it may be interesting for further study.…”