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[Regular tetration] [Iteration series] norming fixpoint-dependencies
If we do regular tetration having a "nice" base 1<b<eta we have two real fixpoints, fp0,fp1.
So, for base b=sqrt(2) we have fp0=2 and fp1=4 .

Iterations on the real axis constitute -at a first sight- three different segments
.            seg1         seg2          seg3
.      --------------|-------------|-------------
.        (0)  ..   (fp0)   ..    (fp1)   ..    (+oo)
.        (0)  ..    (2)    ..     (4)    ..    (+oo)     // for instance for base sqrt(2)
.      --------------|-------------|-------------
.         x1->y1         x2->y2       x3->y3
For some x1 of seg1 we may iterate with infinite height to some y1, but y1 will remain in seg1. However, negative heights may result in imaginary results or even in the log(0)-singularity, so I didn't include that range in the sketch above.

For some x2 in seg2 we can iterate with arbitrary heights to some y2, but again are confined to y2 in seg2.

For some x3 in seg3 we can iterate with arbitrary negative height to some y3 and -in principle- also to arbitrary positive heigt, but practically encounter numerical overflow very soon.

The powerseries for regular tetration can be developed around fp0 or fp1. Let's call that tet0 and tet1 for shortness.

Then if we look at the schroeder-function of tet0 for all x1 we get negative values and for all x2 we get positive values. Thus we can map the set of seg1 to that of seg2 by negating the schroeder-value. This means for instance, that x1 = 1 gets mapped to x2 = 2.467914... and because the height-limits of seg2 are -infinity and +infinity we could use that value x2 to define a norm for that segment, so in seg2 the value x2=2.467914 could be said has (real) height 0 by definition.

We have a similar problem with seg3: here too we have infinity at both height-limits. But we can repeat the norming-process, now using the schroeder-values of tet1. We compute the schroeder-value of x2 using tet1 and compute x3 by negating that value. However, without further measures we get infinity here.
If we reduce the height of x2 by 2, then we get x3 = 417.234406762

So we have the segments with the normed heights
.            seg1                   seg2                 seg3
.     --------------------|-----------------------|-------------
.             .. 1 ..   (fp0)   ..              (fp1)   ..            (+oo)
.             .. 1 ..    (2)    .. 2.467 ...     (4)    .. 417 ...    (+oo)     // b=sqrt(2)
.      -------------------|-----------------------|-------------
.              h(x1)=0  oo|oo     h(x2)=0      -oo|***************    // set norms for tet0
.        *****************|oo     h(x2)=0      -oo|-oo  h(x3)=?=-2    // set norms for tet1

Unfortunately this has two asymmetries: the tet0 and tet1 have somehow opposite sign; but more inconvenient is, that we cannot have the same height-norm.

What we can do is to shift left and use x1=0 as reference. We get then

.         x1 = 0     x2 = 2.606584    (x3=417.2344)
.            seg1                   seg2                 seg3
.     --------------------|-----------------------|-------------
.        (0)  .. 0 ..   (fp0)   ..              (fp1)   ..            (+oo)
.        (0)  .. 0 ..    (2)    .. 2.606 ...     (4)    .. 417 ...    (+oo)     // b=sqrt(2)
.      -------------------|-----------------------|-------------
.             h0(x1)=-1 oo|oo    h0(x2)=-1     -oo|***************    // set norms for tet0
.        *****************|oo    h1(x2)=-1     -oo|-oo h1(x3)=?=-2    // set norms for tet1

and still x3 computed by x2 seems to become infinite. We may reduce again x2 by height 1 to get the usable x3-value of 417.2344...
We cannot reduce x1 by one more height, but my proposal here is to use x1 = b^^-1.5 as reference value.
Then we have, for base b=sqrt(2) the reference-values for height -1.5
.       x1 = -1.33729937324   x2 =  2.68345013524     x3 =  3465302.30778          
.            seg1                   seg2                 seg3
.     -----------------------|-----------------------|-------------
.             ..      ..   (fp0)   ..              (fp1)   ..             (+oo)
.             ..-1.33 ..    (2)    .. 2.683 ...     (4)    .. 3465302. ...(+oo)     // b=sqrt(2)
.      ----------------------|-----------------------|-------------
.           h0(x1)=-1.5    oo|oo    h0(x2)=-1.5   -oo|***************    // set norms for tet0
.          ******************|oo    h1(x2)=-1.5   -oo|-oo h1(x3)=-1.5    // set norms for tet1

The inversion of sign of the schroeder-function-value is essentially the iteration with an imaginary height. For notation I introduce now u0 = ln(fp0) and u1 = ln(fp1)

If we have, for some x, the schroeder-value s, then the schroeder-value of the h'th iterate of x is s*u^h and the negation of sign can be achieved by supplying the according complex value in h.
Using the different fixpoints and different u0 and u1 we can state this norming more explicitely
.        x1 = tet0(1,   -1.5)    
.        x2 = tet0(x1, Pi*I/ln(u0))
.        x3 = tet1(x2, Pi*I/ln(u1))
which define the heights -1.5 for the two tetrations in the three segments.

What is now interesting is, whether the observed wobbling of the tetrates in seg2 using the different fixpoints changes in some interesting way. I remember that the shifting of the height by a half-unit made some significant change in the wobbling when I considered the infinite alternating iteration series (tetra-series) in one of my older msgs, I'll have a look at it soon.

Gottfried Helms, Kassel

Messages In This Thread
[Regular tetration] [Iteration series] norming fixpoint-dependencies - by Gottfried - 07/28/2010, 03:04 PM

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