Re: How can this tetration be extended?

On Jun 23, 9:12 am, "I.N. Galidakis" <morph...@xxxxxxxxxxxx> wrote:
mike3 wrote:
Hi.

I've seen and played with various candidates for an extension of the
tetration (power tower, hyper4, "repeated exponentiation") function to
real towers (the number of exponents.). Most of them focus on real-
number bases, often the ranges 1 < b < e^(1/e), b = e^(1/e), and b > e^
(1/e) (or b > 1). But what about b in the range from 0 to 1? The range
e^-e < b < 1 converges similar to 1 < b < e^(1/e) but oscillates as it
does so. The "regular iteration" and "matrix operator" (Bell/Carleman
matrix power) gives complex values for this tetration at fractional
"tower", and when graphed on a complex plane, it forms a spiral as the
tower is increased continuously from 0, that homes in on the fixed
point. e^-e also appears to converge but at a slower rate.

But what about 0 < b < e^-e? Here it also oscillates but does NOT
converge to a fixed point (it has a real, but repulsing, fixed point
so regular iteration won't work, for example.). I'm also thinking it
"should" have complex values at fractional towers,

Why "should" it have complex values at fractional powers? How did you come to
this conclusion?


Well for one thing, if you look at a graph of say, the regular
iteration
iterating the convergent region at a fractional "height", i.e. a graph
of ^0.5 z for z in that region, you'll see that complex values are
obtained
for e^-e < b < 1, as I mentioned, and there is a branch cut (when you
use
principal branch of logarithm) that goes from 1 along the real line, z
< 1.
I don't see why this pattern should suddenly stop at e^-e, so it would
seem
reasonable to expect complex values for anything that would expand
upon the
values obtained via this method, in that range. Some discussion on
this
topic:

http://math.eretrandre.org/tetrationforum/showthread.php?tid=87&page=1

but has anyone
tried to come up with an extension to them? If so, what may a graph of
the real/imag. parts of the tetration look like (as the tower is
increased continuously from 0), or a parametric one (like what reveals
the "spiral" for the e^-e < b < 1 with the extensions mentioned)? Say
for b = 0.05.

As you say, for 0 < b < e^(-e) the iterates of the power-tower z^^n suffer a
Hopf bifurcation, being a 2-cycle, with the bifurcation point being e^(-e).
Necessarily then, any extension which wants to call itself a "tetration
extension", must interpolate between the extreme points of that cycle.

If the extension is smooth, then the curve should be smooth, too. My own
extension (linear) interpolates between the points of the cycle, but the curve
is not differentiable exactly at the points of the cycle. It's just an
alternating curve, like sin.

You might want to check with the Tetration Forum and see how Andrew Robbins'
extension behaves in that interval. My guess is that it interpolates between the
points of the cycle in a C^{oo}  way.


Robbins' doesn't seem to be stable there or might otherwise have some
caveat. I remember him mentioning something about that, I think.

A semi-interesting consequence of that, would be that if we define 0^^n = 1 for
n even and 0 for n odd, any tetration extension should interpolate there between
the values 0 and 1.

I'm betting 0 is some sort of singularity and branch point too -- it
certainly is
for ^2 z, ^3 z, ^4 z, ^5 z, ... So I'm not sure if ^z 0 should be
defined for
noninteger z, at least from a "complex" point of view.

I suppose from a "real analysis" view though (and if you don't want to
have
complex-valued at fractional tower) we could make ^x 0 a square wave
that's 1 for
x in [0, 1), 0 for x in [1, 2), 1 again for x in [2, 3), 0 again for x
in [3, 4),
1 yet again for x in [5, 6), etc. Certainly any function that is real
valued
and crammed into [0, 1], will be immediately transformed to this type
of alternating
square wave when 0^x is applied to expand it over the rest of the
line, via the
formula f(x+1) = 0^f(x).
.

Relevant Pages

  • Re: How can this tetration be extended?
    ... tetration (power tower, hyper4, "repeated exponentiation") function to ... so regular iteration won't work, ... any extension which wants to call itself a "tetration ...
    (sci.math)
  • Re: How can this tetration be extended?
    ... tetration (power tower, hyper4, "repeated exponentiation") function to ... so regular iteration won't work, ... any extension which wants to call itself a "tetration ...
    (sci.math)
  • Re: Tetration via Newton series -- "finite difference method"
    ... tetration is only defined on integers, ... in setting up an extension to real values of the ... The Newton series about m is defined by ... tetrated to the tower of 1/2), ...
    (sci.math)
  • Re: How can this tetration be extended?
    ... I've seen and played with various candidates for an extension of the ... tetration (power tower, hyper4, "repeated exponentiation") function to ...
    (sci.math)
  • Re: Why is it so difficult?
    ... Why is it so difficult, anyway, to extend tetration to real numbers? ... power, but hard to define 2 tetrated to the one half tower? ... Try doing the same with tetration. ...
    (sci.math)