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Results tagged with complex-dynamics
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user 128556
Dynamics of holomorphic transformations; Mandelbrot and Julia sets.
4
votes
Accepted
Computing the maximum modulus
A partial solution: As mentioned by @MargaretFriedland, the desired $M_a(r)$ is the absolute maximum of $g(t):=\sqrt{{\rm{e}}^{2r\cos t}+2a{\rm{e}}^{r\cos t}\cos(r\sin t)+a^2}$. Notice that this is a …
- 3,599
12
votes
Connected set in a filled Julia set
Not if the Julia set is an interval. Consider the Chebyshev polynomial $f(z)=z^2-2$ for example whose Julia set/filled Julia set is given by $[-2,2]$. The only connected subset containing Julia points …
- 3,599
1
vote
Fully invariant measures for rational functions
The way that the paper by Freire-Lopes-Mañé makes sense of $f^*\mu=d\mu$ is the following: ''For any Borel subset $A$ of $\Bbb{C}_\infty$ with $f\restriction_A$ injective, one has $\mu(f(A))=d.\mu(A)$ …
- 3,599
5
votes
Accepted
Hausdorff dimension of Julia set
This argument is from Eremenko-Lyubich survey: Let $f$ be a rational map and $\mu$ an $f$-invariant ergodic measure on $\widehat{\Bbb{C}}$. By Ledrappier-Young entropy formula, $h_\mu(f)$ is the produ …
- 3,599