Continuity of Julia Sets Under Polynomial Degeneration

Definitions and mathematical statement

For a complex polynomial $f$ of degree at least two, define its filled Julia set and Julia set by \[ K(f)=\{z\in{\mathbb{C}}:\sup_{m\geq0}|f^{\circ m}(z)|<\infty\}, \qquad J(f)=\partial K(f). \] These are nonempty compact subsets of ${\mathbb{C}}$. Write \[ f_j(z)=\sum_{r=0}^d a_{r,j}z^r,\quad a_{d,j}\ne0,\qquad a_{r,j}\to a_r,\quad f(z)=\sum_{r=0}^d a_rz^r. \] The source fixes the degree $d$ of $f_j$, but does not explicitly require $d\geq2$ or $a_d\ne0$ for the limit. These conditions are necessary to use the usual nondegenerate compact-set formulation without further conventions.

For nonempty compact $A,B\subseteq{\mathbb{C}}$, put \[ d^+(A,B)=\sup_{a\in A}\inf_{b\in B}|a-b|, \qquad d_H(A,B)=\max\{d^+(A,B),d^+(B,A)\}. \] Upper semicontinuity of $f\mapsto K(f)$ in the nondegenerate degree-$d$ parameter space means \[ d^+(K(f_j),K(f))\longrightarrow0, \] while lower semicontinuity of $f\mapsto J(f)$ means \[ d^+(J(f),J(f_j))\longrightarrow0. \] Hausdorff continuity requires both directional distances to tend to zero for the relevant set-valued map. The question asks which of these or other compact-set convergence statements hold and under which dynamical hypotheses, rather than assuming unrestricted Hausdorff convergence.

If degree drop is allowed, a different formulation is required. For example, $f_j(z)=z^2/j\to0$ coefficientwise has $K(f_j)=\{|z|\leq j\}$ and $J(f_j)=\{|z|=j\}$, while the bounded-orbit set of the constant limit is all of ${\mathbb{C}}$. Thus ordinary nonempty-compact Hausdorff convergence in ${\mathbb{C}}$ no longer has the same target. A spherical or local set topology, or exclusion of degree drop, cannot be silently chosen.

The supplied assessment dated 2026-08-21 attributes to Douady upper semicontinuity for filled Julia sets, lower semicontinuity for Julia sets, and continuity under a precise dynamical condition, while citing parabolic implosion as an obstruction to unrestricted continuity. The condition and theorem scope are not reproduced; in particular this summary is not extended to the degree-drop situation.

Short English statement

The question asks how Julia and filled Julia sets vary under coefficient convergence, distinguishing one-sided semicontinuity from Hausdorff continuity. The usual fixed-degree theory requires the limit to retain degree at least two; the source wording leaves that point open, and degree drop changes the compact-set setting.

Sources

• Local UnsolvedMath registry, permanent ID $30000944$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Arbeitsgemeinschaft: Julia Sets of Positive Measure (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/17, pp. 869--906. https://doi.org/10.4171/owr/2008/17.
• Supplied reference: Douady, "continuity of Julia sets," https://doi.org/10.5802/xups.1994-03.
• The supplied corrected_verified note reports checking content against the official report. The partially solved label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of exact dynamical hypotheses and status.