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Results tagged with complex-dynamics
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user 3993
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
5
votes
Clustering of periodic points for a polynomial iteration of $\mathbb{C}$
While Robert Israel's answer is correct, $x^2-2$ is a very exceptional case.
In a way, how close do they get is not the interesting question: rather, what gets strange is how far apart do the closest …
- 11.8k
38
votes
Accepted
When does iterating $z \mapsto z^2 + c$ have an exact solution?
No, there are no others.
Analytically, one can show that if a Julia set contains an analytic arc, it is in fact a straight line or a circle (up to conjugation). For the class $z^2+c$, $0$ and $-2$ …
- 11.8k
6
votes
Symmetries of the Julia sets for $z^2+c$
Many Julia sets are known to be quasi-self-similar. This means that there is a quasi-conformal map (thus of bounded distortion) which maps parts of the Julia set to the whole. In fact, given any com …
- 11.8k
1
vote
Accepted
Help determining the asymptotic behavior of an integral involving rational functions.
If I understand your problem correctly, i.e. that $\mu$ is the usual metric on the Riemann sphere, then you're asking if essentially all orbits are expansive, at least with respect to that metric.
Th …
- 11.8k
1
vote
Dense orbits for a rational map
Even for polynomials, the answer is No: See the question on Smooth Julia Sets for links to papers with more details.
Note that for rational functions, there is an additional case, as sometimes the Jul …
- 11.8k
7
votes
Who proved that the Mandelbrot set's Julia sets are locally connected?
If you mean
are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively).
are the …
- 11.8k