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Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps

  • * Corresponding author: Youming Wang

    * Corresponding author: Youming Wang 

This work is supported by the NSFC (grant Nos. 11671092, 11671191, 11871208), by the Scientific Research Foundation of Hunan Provincial Education Department (grant No. 16C0763), by Natural Science Foundation of Hunan Province (grant No. 2018JJ2159) and by the Fundamental Research Funds for the Central Universities (grant Nos. 0203-14380022 and 0203-14380025)

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  • In this paper, we investigate the dynamics of the following family of rational maps

    $ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $

    with one parameter $ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $, where $ n\geq 2 $. This family of rational maps is viewed as a singular perturbation of the bi-critical map $ P_{-n}(z) = z^{-n} $ if $ \lambda \neq 0 $ is small. It is proved that the Julia set $ J(f_\lambda) $ is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of $ f_\lambda $ are attracted by the super-attracting cycle $ 0\leftrightarrow\infty $. Furthermore, we prove that there exists suitable $ \lambda $ such that $ J(f_\lambda) $ is a Cantor set of circles but the dynamics of $ f_{\lambda} $ on $ J(f_{\lambda}) $ is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of $ J(f_{\lambda}) $ is also proved if the free critical orbits are not attracted by the cycle $ 0\leftrightarrow\infty $. Finally we give an estimate of the Hausdorff dimension of the Julia set of $ f_\lambda $ in some special cases.

    Mathematics Subject Classification: Primary: 37F45; Secondary: 37F10, 37F25.


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  • Figure 1.  The Julia sets of $ f_\lambda $ for different $ \lambda $'s when $ n = 4 $. Top left: $ \lambda = 0.8 + 0.3 \rm{i} $ and $ J(f_\lambda) $ is a quasicircle; Top right: $ \lambda = 0.4 $ and $ J(f_\lambda) $ is a Cantor set of circles; Bottom left: $ \lambda = 0.7 $ and $ J(f_\lambda) $ is a Sierpiński carpet; Bottom right: $ \lambda = 0.92 + 0.01 \rm{i} $ and $ J(f_\lambda) $ is a degenerate Sierpiński carpet

    Figure 2.  The non-escaping loci of $ f_\lambda $, where $ n = 3 $ and $ 4 $. Left: $ n = 3 $, the McMullen domain does not exist and the Julia set $ J(f_\lambda) $ cannot be a Cantor set of circles; Right: $ n = 4 $, there is a punctured domain centered at origin which corresponds to the McMullen domain (the big white part in the center)

    Figure 3.  The above and below pictures illustrate the mapping relations of $ h_\lambda $ (see (1)) and $ f_\lambda $ respectively when $ D_0 $ contains one of the free critical values but contains no free critical points. One can observe clearly that $ f_{\lambda} $ and $ h_{\lambda} $ are not topologically conjugate on their corresponding Julia sets

    Figure 4.  The Julia sets of $ f_\lambda $ with $ n = 4 $, $ \lambda = 0.4 $ and $ F(z) = z^3 + 0.01/z^3 $. Both of them are Cantor circles. But $ f_{\lambda} $ and $ F $ are not topologically conjugate on their corresponding Julia sets

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