Conference rusure::math

    This is a hyperbolic fractional linear transformation of the complex
    plane with fixed points at (7 +/- sqrt(5))/2, about 2.38197 and 4.61803.

    By a suitable transformation, these can be sent to zero and infinity
    respectively, when the action of the transformation is a simple
    scaling by a real number larger than 1.

    Thus, any point not equal to 2.38197... will be attracted to the
    other fixed point.

    - Jim

    Eric playing with his calculator again...

If I understand your "thus" sentence, you're saying that

	7 - 11/n

is associated with the TWO numbers (7�sqrt(5))/2, and that any number we
choose for n other than (7-sqrt(5))/2 will produce values which when plugged
back in produce values migrating towards (7+sqrt(5))/2.

Why such assymetry ?  I would expect that some n's would tend towards one
number, and some towards the other.  Why do most numbers migrate only to
the LARGER value ?

/Eric-using-math-as-his-drug-of-choice

>    Thus, any point not equal to 2.38197... will be attracted to the
>    other fixed point.

Well, points like 0, 11/7, 77/38, 418/189, ... may be "attracted" to
that fixed point, but there's an obstacle in the way.  :^)

           <<< Note 1351.4 by TRACE::GILBERT "Ownership Obligates" >>>

>    Thus, any point not equal to 2.38197... will be attracted to the
>    other fixed point.

>Well, points like 0, 11/7, 77/38, 418/189, ... may be "attracted" to
>that fixed point, but there's an obstacle in the way.  :^)

    Nice try, but you're wrong.  Where does that "obstacle" get mapped to
    the next iteration?

    Remember that the point at infinity is just as "nice" as any other
    point as far as fractional linear transformations are concerned.

    - Jim

    Re .3

    In general, consider the conformal automorphisms of the Riemann sphere

	    a z + b
	w = -------
	    c z + d

    This will have one or two fixed points by solving the quadratic

	    a z + b
	z = -------
	    c z + d

    Suppose the fixed points are distinct, say p and q.

    Then by a change of variables we can send q to zero and p to infinity,
    where the transformation takes the simple form

	w' = k z'


	     w - q	     z - q
        w' = -----,	z' = -----
	     w - p	     z - p


    In your case, k is a real constant greater than zero.  Does the assymetry
    make sense now?

    If k is real the transformation is hyperbolic and the orbits flow from
    one pole to the other like longitude lines.  If k is complex with
    magnitude 1, the transformation is elliptic and the orbits are latitude
    circles.  Otherwise the orbits are loxodromic curves on the Riemann
    sphere (like the loxodromes from navigation.)

    If p = q, the transformation is parabolic; sending the fixed point to
    infinity the transformation is equivalent to a translation w' = z' + k.

    Another way to see this is to look at nonsingular 2 by 2 complex
    matrices modulo scalar multiples of the identity: the projective
    general linear group PGL(2,C).  Then the fixed points correspond to
    the eigenvectors.

    Yet another geometric point of view is to look at the projective
    transformations of 3 space that respect the unit sphere.  These are
    those 4 by 4 matrices common to 3D computer graphics that satisfy a kind
    of orthogonality, preserving the form x^2 + y^2 + z^2 - w^2 = 0,
    the homogenous equation of the sphere.

    These transformations send planes to planes, and planes cut the sphere in
    circles.  There is an isomorphism between this class of 4 by 4 real
    matrices and the 2 by 2 complex matrices above which you can establish
    by stereographic projection of the complex plane to the sphere.

    - Jim

    7 - 11/x = 7 - 11/(7-11/x) ==>
    
    1/x = x/(7x-11) ==>
    
    x^2 - 7x + 11 = 0 ==>
    
         7 + sqrt(5)            7 - sqrt(5)
    x = -------------   or x = ------------------
              2                      5
    
    Hence, the limits if they exist.
    
    Eugene