Locally connected models for Julia sets

Let $P$ be a polynomial with a connected Julia set $J$. We use continuum theory to show that it admits a \emph{finest monotone map $\ph$ onto a locally connected continuum $J_{\sim_P}$}, i.e. a monotone map $\ph:J\to J_{\sim_P}$ such that for any other monotone map $\psi:J\to J’$ there exists a monotone map $h$ with $\psi=h\circ \ph$. Then we extend $\ph$ onto the complex plane $\C$ (keeping the same notation) and show that $\ph$ monotonically semiconjugates $P|_{\C}$ to a \emph{topological polynomial $g:\C\to \C$}. If $P$ does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi’s fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a criterion and a useful sufficient condition for the map $\ph$ not to collapse $J$ into a point.

💡 Research Summary

The paper addresses a fundamental problem in complex polynomial dynamics: how to obtain a locally connected model of a Julia set that is connected, without relying on heavy analytic machinery. For a polynomial (P) whose Julia set (J) is connected, the authors employ tools from continuum theory to construct a canonical monotone map
\