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 [UFO] - a contradiction in assuming continuous tetration? Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/23/2010, 10:53 AM (This post was last modified: 08/30/2010, 05:03 AM by Gottfried.) Hmm; it seems impossible that the following was not yet discussed in our earliest tetration-discussion, but I can't remember and don't have an idea, where in our threads I possibly could find this. It just occured to me yesterday in a bar nearby where I tried to vizualize the spiral form of trajectory for integer-height-tetration in terms of the log-polar-representation on an envelope, with a beer at side... 1) having a real base, say b=exp(1), we assume, that beginning at a real value for x0 some positive integer or real iteration-height h gives some value x1, again on the real line. So, if we fix some point x0 = 1.2 then for the continuous interval for h, say h=1..2 we get a continuous interval for x1, say x1=3.32 .. 27.66 . Or if we take x0=-4, then for h=1..2 we assume the continuous real interval for x1 of about x1=0.018 ... 1.0184 2) For complex values x0 the trajectory of the integer-height iterates x1,x2,... can have the shape of a spiral. Intuitively we would assume, that fractional height iterates for a continuous interval of heights h=1..2 fill that shape with a continuous curve, which roughly follows that spiraling shape. 3) There are complex values in the first quadrant, whose integer height iterations occurs in the second, and also in the third and forth quadrant. For instance [h=-1 x_=1+0.5*î // update: I forgot in the first writing of this msg that the mysterious x0 below was in fact x1 iterated from that x_ , sorry] h=0 x0 = 2.38551673096 + 1.30321372969*I h=1 x1 = 2.87262925108 + 10.4780327918*I = exp(x0) h=2 x2 =-8.74880093578 - 15.3675936421*I = exp(x1) By (2) we expect, that the positive-real iteration curve from x0 to x2 is continuous and thus must cross the real axis. This happens for some fractional height h=1+µ ; so there exists some purely real iterate x_h when we begin at x0 and iterate continuously using base e. Also iterating further continuously with real heights we arrive at x2, which is a complex value (in the thirs quadrant). But this contradicts (1), where we assume, that a positive real-height iterate of a real value x0 leads to a real value x1. [arrgggh] The evening is gone, the morning has come...still I've no other idea than that of a mindbreaking contradiction... Gottfried Helms, Kassel mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 08/23/2010, 09:08 PM (This post was last modified: 08/23/2010, 09:29 PM by mike3.) The problem is that on the complex plane, the continuum iteration is multivalued. You have to remember that, say, $\exp^{1/2}(z)$ is a multivalued "function" in the same sense that $\log(z) = \exp^{-1}(z)$. The equation $\exp^u(\exp^v(z)) = \exp^{u+v}(z)$ only holds for general real (or complex) $u$, $v$, and complex $z$ if one chooses the correct branches of the functions involved for the given set of parameters. Failure of this identity to hold (which you assume when iterating "further" from a value obtained by one continuum iteration) is no different than the fact the equation $\log(\exp(z)) = z$ does not hold for all $z$. Let's examine this iteration process more closely. Note that $\exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z))$, and both $\mathrm{tet}$ and $\mathrm{slog}$ are multivalued functions. Consider taking the iteration of that value at $1 + \mu$. We have $\exp^{1 + \mu}(x_0) = \mathrm{tet}(1 + \mu + \mathrm{slog}(x_0))$. Now consider iterating that a little more, say, $\delta$. We would want to do this: $\exp^{\delta}(\exp^{1 + \mu}(x_0)) = \exp^{\delta + 1 + \mu}(x_0)$, which, in detail, means we'd like to try doing this: \begin{align} \exp^{\delta}(\exp^{1 + \mu}(x_0)) &= \mathrm{tet}(\delta + \mathrm{slog}(\mathrm{tet}(1 + \mu + \mathrm{slog}(x_0)))) \\ &= \mathrm{tet}(\delta + 1 + \mu + \mathrm{slog}(x_0)) \\ &= \exp^{\delta + 1 + \mu}(x_0)\\ \end{align} But if we mull over these steps, we see that the second equality cannot be justified. We cannot necessarily say that $\mathrm{slog}(\mathrm{tet}(z)) = z$ for a general complex $z$ any more than we can say $\log(\exp(z)) = z$ for a general complex $z$. It's all because of the ambiguity of the multivalued functions involved. Even the single-valued principal branch of $\mathrm{tet}$ is not an injective function on the complex $z$-plane, as can be seen by inspecting its graph. Indeed, this paradox shows that no matter how we may try to extend $\mathrm{tet}$ to the $z$-plane, we cannot make it injective, at least if our $\mathrm{tet}$ is continuous (up to a cut, anyway). Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/23/2010, 10:03 PM (This post was last modified: 08/23/2010, 10:05 PM by Gottfried.) (08/23/2010, 09:08 PM)mike3 Wrote: The problem is that on the complex plane, the continuum iteration is multivalued. (...) Indeed, this paradox shows that no matter how we may try to extend $\mathrm{tet}$ to the $z$-plane, we cannot make it injective, at least if our $\mathrm{tet}$ is continuous (up to a cut, anyway). Hi Mike - yes, I think the multivaluedness of the log propagates to the slog, and that this gives problems for the tetrate in the complex plane. However I can't follow completely. The multivaluedness of the log does not imply, that at each point z the log(exp(z)) is arbitrary; it is multivalued, but the different values are distinct. If we look at a small delta-region around z, the images of log(exp(z+delta)) are continuous around each of the multiple values of log(exp(z)), isn't it? (I mean except of the cut-line). I think, the multivaluedness gives continuous orbits but on distinct pathes, and not, say, continuous "smeared regions" of arbitrary change of direction for some continuous real delta-height. Hmm - I've near null experience in discussion of such matter, so please bear with me if I'm wrong here or expressed myself unclear. Gottfried Gottfried Helms, Kassel mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 08/24/2010, 01:39 AM (This post was last modified: 08/24/2010, 02:51 AM by mike3.) (08/23/2010, 10:03 PM)Gottfried Wrote: (08/23/2010, 09:08 PM)mike3 Wrote: The problem is that on the complex plane, the continuum iteration is multivalued. (...) Indeed, this paradox shows that no matter how we may try to extend $\mathrm{tet}$ to the $z$-plane, we cannot make it injective, at least if our $\mathrm{tet}$ is continuous (up to a cut, anyway). Hi Mike - yes, I think the multivaluedness of the log propagates to the slog, and that this gives problems for the tetrate in the complex plane. However I can't follow completely. The multivaluedness of the log does not imply, that at each point z the log(exp(z)) is arbitrary; it is multivalued, but the different values are distinct. If we look at a small delta-region around z, the images of log(exp(z+delta)) are continuous around each of the multiple values of log(exp(z)), isn't it? (I mean except of the cut-line). I think, the multivaluedness gives continuous orbits but on distinct pathes, and not, say, continuous "smeared regions" of arbitrary change of direction for some continuous real delta-height. Hmm - I've near null experience in discussion of such matter, so please bear with me if I'm wrong here or expressed myself unclear. Gottfried Not arbitrary, ambiguous, in that there isn't a single answer. What's arbitrary is the choice of a certain specific "principal value" for tetration/slog (or any other multivalued functions for that matter.). This explains what is going on. The iteration of exp, $\exp^\delta$ for non-integer real $\delta$, that we apply upon reaching the real number $z_{1 + \mu} = \exp^{1 + \mu}(z_0)$, is "ambiguous" (as was $\exp^{1 + \mu}(z_0)$ itself): the path walking it along the real line is not the only valid one -- the path on the remaining part of the "spiral" is another, and this is what resolves your paradox. To get it, you just use a different branch of $\exp^\delta$, namely the one for which $\mathrm{slog}(z_{1 + \mu}) = 1 + \mu + \mathrm{slog}_{[0]}(z_0)$ (with $\mathrm{slog}_{[0]}$ being the principal branch of $\mathrm{slog}$.). And no, it doesn't give a continuous smear, but a countably infinite discrete set of "valid" paths (there's a proof that any Riemann-multifunction can only have countably many values). These may or may not be dense in the plane, I don't know. EDIT: Here's a graph which shows the principal path, using the Cauchy-integral tetrational (presumably also equivalent to the Kneser and "intuitive" Abel matrix tetrationals). It is obvious that a branch change is required at some point(s) along the way to keep following it. Numerical testing suggests that the failure occurs at $h \approx 1.492$, and this is where we must leave the principal branch of $\mathrm{slog}$. Note it happens shortly before the path hits the real axis. This means further real-height iteration will not move us along the real axis for that point that is on it. There are additional points where we must change branch again and again all along the path, but I currently don't have a good way to access arbitrary branches of $\mathrm{slog}$ (it has a very complex, nested structure. Tetration is much more complicated than exponentiation, as you may have noticed here!).     Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/24/2010, 04:25 AM (This post was last modified: 08/24/2010, 09:59 AM by Gottfried.) Well, I didn't get the regular tetration for complex fixpoints properly working until now, but because you show that picture I'll give it a new try. Surprise, surprise... I got the regular tetration for base exp(1) working, also for fractional heights... simply when I applied my newer routines to the old stuff... What I did was to use the integer-height tetrate z1 = itet(z0,-10) of height -10 (where z0 was taken from x0 in my initial post); computed via the schroeder-function and its inverse the fractional values in the interval of h=0..2 and lifted the results by integer-height tetration of +10. With this I got the graph below. Now: in my computation I had no situation where I had to change the branch for logarithm. What's going on here? I understood your msg such, that you had to change the branch in the near of h=1.49 ? Or did you mean, one *had to* change the branch *if* one wanted to proceed on the real axis alone (at h=1.49** )? Besides, in my regular tetration version now the real point is at height 1.498+eps - does this conform with your value? Gottfried [Update2]: Well, just it occurs to me, that at h1~1.43 and h1~1.93 ~ h1+0.5 the curve crosses itself. After some binary search I find, that the height-difference is not exactly 0.5, so this does not indicate a "half-iterate-fixpoint", but something possibly with some irrational period? The point is in the height-intervals h1 in 1.4371584...1.4372544 and h2 in 1.94060196...1.94062644 . The point itself is in the area [ -30.6552101206 + 12.4460532739*I, -30.6707389330 + 12.4324857727*I ; -30.6634224893 + 12.4493356383*I, -30.6641828863 + 12.4300037481*I] [/update2] Appendix: The first picture is the overview using z0=2.38551673096 + 1.30321372969*I for h=0..2 in steps of delta-h=0.01     The second picture shows the detail at h=1.49..1.5 where the curve crosses the real axis. By binary-search I get h_r=1.49800142290, z_r = -39.4162061931 + 0*î     Gottfried Gottfried Helms, Kassel tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 08/24/2010, 12:17 PM this seems exactly the same thing i was talking about in my thread " peroid of exp exp exp " every z_0 has its period. and those periods form a periodic function themselves. i didnt get any replies to that thread ... maybe im wrong , or maybe it wasnt clear that it was related ?? Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/24/2010, 02:29 PM Ok, I think I found it, where the multivaluedness of log comes into the play. For the following assume the notations: e = exp(1) b : for base (here b=e ) t : first fixpoint, imag(t)>0 u : ln(t) x' := x/t-1 x" := (x+1)*t For simpliness I misuse the common notation with the double-caret in the following way: y=x^^h : meaning y=exp°h(x) for the current base b=e, so (x^^h)^^k = x^^(h+k) x=yh : meaning x=log°h(y) for the current base b=e So actually t can be approximated by t = limit{h->infty} ( zh ), where z is an appropriate nearly arbitrary initial value. If I set x0 = and x1 = x0^^1 and want to find the fractional iterates in that interval, I look first, whether the powerseries of the schroeder-function schroeder(x') converges for x0' and x1' . If not I use w0 = xh0 for some h0 so that for some eps, for instance eps=1e-4 according to my wished accuracy, we have abs(w0-t) < eps Then I compute the value of the schroeder-function for this w0, say s0=schroeder(w0') Then I compute a set of values S such that S(h) = s0*u^h for h=0..1 . The assumption behind this is, that in the limit S is dense and continuous for the continuous parameter h. Then the inverse Schroeder-function is used for all values S(h) of the set S: T(h) = schroederInv(S(h))" = (schroederInv(S(h)) +1)*t and finally I have to correct for the initial integer-height-reduction by h0: X(h) = T(h)^^h0 Again we assume, that X(h) is in the limit continuous in that interval h=0..1, so that X(h) covers the interval x0^0 .. x0^^1 = x0..exp(x0) , the fractional iterates for one unit-height-interval. Having that interval, the other unit-intervals for higher iteration-heights are then computed by integer-height iteration X(m+h) = X(h)^^m And now the problem of branch-selection becomes visible: if I compute the set X(2+h) this way I get for the chosen initial value x0 some strongly varying imaginary values. If I do Y(2+h) = log(exp(X(2+h))) then the imaginary values of Y(2+h) are truncated modulo 2*Pi, and also that truncation removes "arbitrarily" multiples of 2*Pi. This is done by the exp-function, and thus the inversion by log cannot recover that integer multiples of 2*Pi. Hmmm. Here I am reminded of my discussion of the tetration in log-polar-representation of complex numbers and the concept of a "winding-number" which should keep track of the integer multiples of 2*Pi when exponentiation takes place... Also this suggests, that for a initial set X computed by fractional iterations of some x0 we must choose x0 and x1 such, that all X(h) for h=0..1 do not exceed 2*Pi*I in their imaginary component... -------- Ah, all this seems to make things clearer now. (Perhaps I'm also going to understand the Cauchy-integral method this way) And the special good news for me is, that I can now extend my Pari/GP-procedures for regular tetration also for bases having a complex fixpoint. At least some more partial solutions. That this was not possible before has really been frustrating... Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/24/2010, 04:30 PM (This post was last modified: 08/24/2010, 04:35 PM by Gottfried.) Here I add four more pictures. The first is the same as the first in my older msg; only that I remembered , that the original x0 was x0=1+0.5*I and the cryptic x0=2.38+1.303*I was in fact exp(x0) = x0^^1, so all height-references are increased by 1. In the next we see the enormous spiraling for one unit-height-interval h=3..4 only! (Don't know whether the regular-tetration does artifacts here, however I checked this with 400 digits float precision) The third and fourth pictures are simply scaled versions of the pictures 1 and 2. Gottfried P.s. the self-crossings of the curve are really impressive! Do I really want a function like this...? 1)     2)     3)     4)     Gottfried Helms, Kassel mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 08/24/2010, 07:19 PM (This post was last modified: 08/24/2010, 07:34 PM by mike3.) First off, the graph I give is not regular, it is the Cauchy-integral tetrational (which *may* be equivalent to Kneser's tetrational and the intuitive Abel-matrix one.). This tetrational is real at the real axis and decays to the conjugate fixed points toward $\pm i \infty$. It is, however, asymptotic to regular tetrationals in the imaginary direction (but two different ones, for the two smallest-magnitude conjugate fixed points). Though your graph from the regular looks sort of like mine. (I presume you used the positive-imag-part conjugate fixed point, which would be roughly like the upper half of the Cauchy-integral tetrational.) The "change of branch" I had been talking about at $h \approx 1.492$ is not of $\log$ but $\mathrm{slog}$. It refers to the point at which in order to advance along the path by $\delta$, i.e. to compute $\exp^{\delta}(\exp^{\alpha}(z))$ and get $\exp^{\delta + \alpha}(z)$, you need to use a different branch of $\mathrm{slog}$ in $\exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z))$, which corresponds to using a different branch of $\exp^\delta$. That there exist paths which self-cross is unavoidable, since the function cannot be injective and I believe must take every complex value (except, perhaps, 0) infinitely many times at infinitely many places. This non-injectivity is the source of the "paradox" you originally mentioned. For the regular this is easy to prove since it is entire and transcendental, then just use Picard's great theorem. Not sure how to prove it for the non-regular Cauchy-tetrational, but the functional equation $F(z+1) = \exp(F(z))$ may be useful ($\exp$ is not injective). EDIT: just saw your second post -- seems you already realized the bit about the functional eq. and how the non-injectivity of $\exp$ contributes. Gottfried Ultimate Fellow Posts: 763 Threads: 118 Joined: Aug 2007 08/24/2010, 08:09 PM (This post was last modified: 08/24/2010, 08:11 PM by Gottfried.) (08/24/2010, 07:19 PM)mike3 Wrote: First off, the graph I give is not regular, it is the Cauchy-integral tetrational (which *may* be equivalent to Kneser's tetrational and the intuitive Abel-matrix one.). This tetrational is real at the real axis and decays to the conjugate fixed points toward $\pm i \infty$. It is, however, asymptotic to regular tetrationals in the imaginary direction (but two different ones, for the two smallest-magnitude conjugate fixed points). Though your graph from the regular looks sort of like mine. (I presume you used the positive-imag-part conjugate fixed point, which would be roughly like the upper half of the Cauchy-integral tetrational.) Hmm - what I see is, that the borders of the plot (the min-max of reals and of imags) differ obviously. Since the x0-value in the introductory msg was truncated to 12 digits the reason for the different plots *could* be such a inaccuracy. As I recalled today in the morning, that value x0 was in fact the value exp(1+0.5*i). Could you compute the min-max-limits of your graph with that exact value again? I'd like to know, whether the difference between the regular and the cauchy-integral-method is really so big. For comparision, I have - real_min~-42.1153427 at height h=2.54 (in new count x=1+0.5*î, h=1.54 for the old count x0=exp(1+0.5*î)), - imag_min~ -58.97 at height h=2.68 , - real_max~ 60.69 at h=2.79 and - imag_max~ 48.4 at h=2.87 (these values are all on my first picture in this thread on the magenta curve) I'll chew the rest of your post later - Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

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