Questions tagged [complex-dynamics]

Dynamics of holomorphic transformations; Mandelbrot and Julia sets.

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Dynamics of $e^z+z^2+z+1$

Question: How to find the dynamics of $f(z)=e^z+z^2+z+1$ also how to find the escaping set ? Since critical points control the dynamics, I want to find the critical points first, but I am not getting ...
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What is the logical complexity of the Mandelbrot Local Connectivity conjecture? (Is it equivalent to a statement of arithmetic?)

Denote by MLC the statement “the Mandelbrot set is locally connected” and MHC the statement “hyperbolic components are dense in the Mandelbrot set” (it is known that MLC implies MHC, and whether ...
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Examples of hyperbolic set and J-stable sets

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
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Does the Mandelbrot set have dense interior?

Let $M$ be the Mandelbrot set. Question: Is the interior of $M$ dense in $M$? My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and ...
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Borel summation and the Abel function of $e^z-1$

This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
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Finding the repelling fixed point of an exponential, knowing only its attracting one

This question has been bugging me for a while, I have an answer that is working sufficiently for the program I'm using, but it is a tad slow, and let's say imprecise. It is not an overtly difficult ...
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On entire functions with polynomial Schwarzian derivative

The Schwarzian derivative of an entire holomorphic function $f$ is defined as $$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$ In the following, we only consider ...
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