Re: Continuous iteration of another quadratic map

On Feb 17, 5:16 pm, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 17.02.2009 21:56 schrieb mike3:> Hi.

I mentioned here about the iteration of the map f(z) = z^2 - 2, which
turned out to have a simple closed form from trigonometric functions.
What about the map f(z) = z^2 - 1? This would not appear to have such
a closed form, however, could there be some sort of power series or
other non-closed-form formula for its general iteration, including to
fractional and real iteration count?

Hm, with

           z0 = 0.5*(1+sqrt(5))  // golden ratio

you get a function without constant term:

     g(z) = x^2 +   2 * z0 * x

which can be used to replace f(z):

     f°h(z) = g°h(z - z0) + z0

for all positive integer heights.

Whether the regular fractional iteration of g(z) is a suitable
interpolation also for f(z) is then "the usual question"...

For f°0.5 (1)  I get 0.587082299302... using g(z-z0) + z0 and the
matrix-method. The powerseries is "friendly"...


Which is between f^0(1) = 1 and f^1(1) = 0, so it would seem
reasonable.

Now comes the really interesting question: would it be possible to
iterate
this continuously on complex z-values, and make a "movie" that shows
how
the graph of the magnitude of the function on the z-plane changes as
the
iteration count ("height", as you call it) is varied from, say, 0 to
30, continuously.
That way one could see a smooth, continuous development of the Julia
fractal it makes. Or even just a plot of the orbit of, say, 0, which
flips
alternately between 0 and -1 on the integers. Is the resulting smooth
function, f^x(0),
a 2-periodic function or is asymptotic to one? A graph of f^x(0) from
x = 0 to x = 10
would be interesting.

Using the "inverse symbolic calculator", the coefficient of x in the
power
series of the hemifunction (g^(1/2)(x)) is an algebraic number equal
to a solution of
x^4 - 2x^2 - 4 = 0, but the coefficient of x^2 could not be found in
the database:
g^(1/2)(x) = (sqrt((2 + sqrt(20))/2))x + <something> x^2 + ...
.

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