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Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much! Edit 1: Suppose further that $f$ is not a Chebychev polynomial and thus the Julia set is not smooth. Edit 2: It is a fact that if $K$ has no interior then its Hausdorff dimension is larger than 1. I wonder whether this might be enough to give a positive answer.

Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much! Edit 1: Suppose further that $f$ is not a Chebyshev polynomial and thus the Julia set is not smooth. Edit 2: It is a fact that if $K$ has no interior then its Hausdorff dimension is larger than 1. I wonder whether this might be enough to give a positive answer.

Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much! Edit: Suppose further that $f$ is not a Chebychev polynomial and thus the Julia set is not smooth.

Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much! Edit 1: Suppose further that $f$ is not a Chebychev polynomial and thus the Julia set is not smooth. Edit 2: It is a fact that if $K$ has no interior then its Hausdorff dimension is larger than 1. I wonder whether this might be enough to give a positive answer.

Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much!

Let $K$ be the filled Julia set of a complex polynomial of degree at least 2. Suppose that $K$ is connected. Let $p_1, \dots, p_N \in K$ be some points. Does there exist a connected set $K_N$ containing $p_1, \dots, p_N$ such that $K_N\subset K$ but $K_N \neq K$? Any references, hints or counterexamples would be very welcome. Thank you very much! Edit: Suppose further that $f$ is not a Chebychev polynomial and thus the Julia set is not smooth.