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August 29, 2022

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In Julia set of a semi-conjugated polynomial, any point is a smooth point?

Consider the Julia set of a semi-conjugated polynomial $P$, where $P$ is post-critically finite. (I will explain it with the example $z^3-2$ later.) $P$ has a critical point at the Julia set. Let $H$ be the local degree of $P$ (or equivalently, the local degree of the restriction $P|_J$ where $J$ is the Julia set of $P$) and $C$ the topological degree of $P$, and $P|_J$ be a semi-conjugated polynomial of the post-critical polynomial $Q$. Then, the degree of $P$ is $HC$ and $H$ is less than $C$.
I have been looking for a proof of the following statement which is not so easy:
“Let $J$ be the Julia set of the polynomial $P$ and $J_k$ be the set of its $k$-th iterate. If $x\in J$, then $x$ is a smooth point in the Julia set.”
However, I couldn’t find a proof of this. Any idea?

It’s true for an arbitrary post-critically finite polynomial, and the proof is almost the same as the proof of the well-known statement for the Mandelbrot set (i.e., for the polynomial $z^2+c$).
Namely, suppose $x\in\mathbb{C}$. We will show that for every $m\in\mathbb{N}$, there exists $r=r(x,m)>0$ such that the $m$-th backward iterate of $x$ lies on the circle $C_r(x)$ of radius $r$ centered at

The Secret Of The Nagas In Hindi.pdf

The Secret Of The Nagas In Hindi.pdf 🤜

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