OFFSET
0,1
COMMENTS
Let x(2) and x(3) denote the remaining zeros of F(x), x(2) < x(3). Then it could be proved that f(x(1)) = x(3), f(x(3)) = x(1), and f(x(2)) = x(2).
We note that the plot of the restriction of F(x) to the interval [-2,2] "is very similar" to the plot of the polynomial (x-x(1))*(x-x(2))*(x-x(3)) for x in [-2,2].
Let A = {x in R: f^n(x) = x(2) for some nonnegative integer n, where f^n denotes the n-th iteration of f}. Then if z is a real number, which does not belong to A, and z(0):= z, z(n+1) = f(z(n)) = sqrt(2)*sin(Pi/4 - z(n)), n in N, then one of the subsequences either {z(2*n-1)} or {z(2*n)} is convergent to x(1) and the second one is convergent to x(3).
LINKS
R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007,745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.
EXAMPLE
We have x(1) = -0.830198517...
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Roman Witula, Sep 19 2012
STATUS
approved