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Results tagged with complex-dynamics
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user 73630
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
4
votes
Dynamics of $e^z+z^2+z+1$
According to the picture below, it seems the Julia set is the whole plane, and escaping set is dense. Darker parts escape faster (and possibly come back by the effect of $e^z$).
$f(z)$ in $[-2\pi,2\pi …
- 106
2
votes
Looking for a precise statement about hyperbolic points in the interior of the Mandelbrot set
It is conjectured that hyperbolic maps are dense in the family of quadratic polynomials $Q_c(z) = z^2+c$.
And it is known
if $c$ is not in the Mandelbrot set, $Q_c$ is hyperbolic.
if $c$ is in the bo …
- 106
0
votes
Why are the Julia sets so simple? (quadratic family)
When you consider infinitely renormalizable polynomial, you can see infinitely many completely different picture. For example, there exists $c$ such that you can find periodic points $x_n$ such that $ …
- 106
1
vote
Accepted
Is it known that MLC is sufficient to prove the density of hyperbolic conjecture of rational...
I guess "MLC implies HD" holds for any unicritical polynomial family (probably the same proof applies but I haven't checked). On the other hand, Lavaurs proved in his thesis that the connectedness loc …
- 106
1
vote
Accessible points of a simply connected domain
Nice references are
Section 17 of Milnor's book "Dynamics in One Complex Variable" (old version is available in arXiv https://arxiv.org/abs/math/9201272 and it is Section 15 for this version), and
Se …
- 106
1
vote
$f(f(z)) = z , f(\exp(z)) = \exp(f(z)) $?
I don't know the complete answer, but here are some examples:
More generally, let $x$ be a repelling periodic point of period $p$ for an entire map $g$. Then there exists a linearizing coordinate $\va …
- 106