Timeline for When does iterating $z \mapsto z^2 + c$ have an exact solution?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Aug 27, 2011 at 22:17 | comment | added | Feldmann Denis | Sorry to contradict you, but there is no reason that a closed form soluiion of the shape $z_n=f(k^nZ_0) (with Z_0=g(z_0)) would "describe" a fractal. See my answer below... | |
| Aug 9, 2010 at 5:44 | comment | added | Victor Protsak | 1. The substitution $z=w+w^{-1}$ that conjugates $z\to z^2-2$ to $w\to w^2$ is the twice the "Zhukovsky function" of complex analysis (which was used for modeling airfoils). 2. The principle behind the Lucas-Lehmer test is an explicit formula for the $n$th iterate of $z\mapsto z^2-2$ that involves raising to the power $2^n.$ 3. There is an algebraic explanation of the special character of $c=2:$ the only monic quadratic polynomials that belong to a family of commuting polynomials of all natural degrees are $z^2$ and $z^2-2$ (for $z^2-2$, the family consists of rescaled Chebyshev polynomials). | |
| Aug 9, 2010 at 4:21 | comment | added | Gerry Myerson | In the Pollard rho method for factoring the integer n, one iterates the map $n\to f(n)$, reducing modulo $n$, for an appropriate $f$, and calculates certain GCDs. It's common to use $f(n)=n^2+c$, and one has to avoid $c=0$ and $c=-2$, for reasons similar to those given in this thread. | |
| Aug 9, 2010 at 0:44 | vote | accept | Richard Borcherds | ||
| Aug 9, 2010 at 0:44 | comment | added | Richard Borcherds | I guess it is unreasonable to expect a completely watertight answer since my question was a bit vague, but looking at the Julia sets does indeed seem to be the key point. I should have looked at en.wikipedia.org/wiki/Julia_set which also points out that 0 and -2 have special Julia sets. It seems I managed to pick at random the only non-zero real number that did not have the property I claimed. | |
| Aug 8, 2010 at 19:26 | history | answered | Jacques Carette | CC BY-SA 2.5 |