[UFO]  a contradiction in assuming continuous tetration?  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: [UFO]  a contradiction in assuming continuous tetration? (/showthread.php?tid=499) Pages:
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[UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/23/2010 Hmm; it seems impossible that the following was not yet discussed in our earliest tetrationdiscussion, 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 integerheighttetration in terms of the logpolarrepresentation 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 iterationheight 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 integerheight 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 positivereal 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 realheight 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... RE: [UFO]  a contradiction in assuming continuous tetration?  mike3  08/23/2010 The problem is that on the complex plane, the continuum iteration is multivalued. You have to remember that, say, is a multivalued "function" in the same sense that . The equation only holds for general real (or complex) , , and complex 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 does not hold for all . Let's examine this iteration process more closely. Note that , and both and are multivalued functions. Consider taking the iteration of that value at . We have . Now consider iterating that a little more, say, . We would want to do this: , which, in detail, means we'd like to try doing this: But if we mull over these steps, we see that the second equality cannot be justified. We cannot necessarily say that for a general complex any more than we can say for a general complex . It's all because of the ambiguity of the multivalued functions involved. Even the singlevalued principal branch of is not an injective function on the complex plane, as can be seen by inspecting its graph. Indeed, this paradox shows that no matter how we may try to extend to the plane, we cannot make it injective, at least if our is continuous (up to a cut, anyway). RE: [UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/23/2010 (08/23/2010, 09:08 PM)mike3 Wrote: The problem is that on the complex plane, the continuum iteration is multivalued. (...) 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 deltaregion 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 cutline). 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 deltaheight. 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 RE: [UFO]  a contradiction in assuming continuous tetration?  mike3  08/24/2010 (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. (...) 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, for noninteger real , that we apply upon reaching the real number , is "ambiguous" (as was 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 , namely the one for which (with being the principal branch of .). And no, it doesn't give a continuous smear, but a countably infinite discrete set of "valid" paths (there's a proof that any Riemannmultifunction 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 Cauchyintegral 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 , and this is where we must leave the principal branch of . Note it happens shortly before the path hits the real axis. This means further realheight 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 (it has a very complex, nested structure. Tetration is much more complicated than exponentiation, as you may have noticed here!). [attachment=734] RE: [UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/24/2010 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 integerheight tetrate z1 = itet(z0,10) of height 10 (where z0 was taken from x0 in my initial post); computed via the schroederfunction and its inverse the fractional values in the interval of h=0..2 and lifted the results by integerheight 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 heightdifference is not exactly 0.5, so this does not indicate a "halfiteratefixpoint", but something possibly with some irrational period? The point is in the heightintervals 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 deltah=0.01 [attachment=735] The second picture shows the detail at h=1.49..1.5 where the curve crosses the real axis. By binarysearch I get h_r=1.49800142290, z_r = 39.4162061931 + 0*î [attachment=736] Gottfried RE: [UFO]  a contradiction in assuming continuous tetration?  tommy1729  08/24/2010 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 ?? RE: [UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/24/2010 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/t1 x" := (x+1)*t For simpliness I misuse the common notation with the doublecaret 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=y``h : meaning x=log°h(y) for the current base b=e So actually t can be approximated by t = limit{h>infty} ( z``h ), where z is an appropriate nearly arbitrary initial value. If I set x0 = <some complex value> and x1 = x0^^1 and want to find the fractional iterates in that interval, I look first, whether the powerseries of the schroederfunction schroeder(x') converges for x0' and x1' . If not I use w0 = x``h0 for some h0 so that for some eps, for instance eps=1e4 according to my wished accuracy, we have abs(w0t) < eps Then I compute the value of the schroederfunction 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 Schroederfunction 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 integerheightreduction 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 unitheightinterval. Having that interval, the other unitintervals for higher iterationheights are then computed by integerheight iteration X(m+h) = X(h)^^m And now the problem of branchselection 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 expfunction, 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 logpolarrepresentation of complex numbers and the concept of a "windingnumber" 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 Cauchyintegral method this way) And the special good news for me is, that I can now extend my Pari/GPprocedures 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 RE: [UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/24/2010 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 heightreferences are increased by 1. In the next we see the enormous spiraling for one unitheightinterval h=3..4 only! (Don't know whether the regulartetration 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 selfcrossings of the curve are really impressive! Do I really want a function like this...? 1) [attachment=737] 2) [attachment=738] 3) [attachment=739] 4) [attachment=740] RE: [UFO]  a contradiction in assuming continuous tetration?  mike3  08/24/2010 First off, the graph I give is not regular, it is the Cauchyintegral tetrational (which *may* be equivalent to Kneser's tetrational and the intuitive Abelmatrix one.). This tetrational is real at the real axis and decays to the conjugate fixed points toward . It is, however, asymptotic to regular tetrationals in the imaginary direction (but two different ones, for the two smallestmagnitude conjugate fixed points). Though your graph from the regular looks sort of like mine. (I presume you used the positiveimagpart conjugate fixed point, which would be roughly like the upper half of the Cauchyintegral tetrational.) The "change of branch" I had been talking about at is not of but . It refers to the point at which in order to advance along the path by , i.e. to compute and get , you need to use a different branch of in , which corresponds to using a different branch of . That there exist paths which selfcross 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 noninjectivity 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 nonregular Cauchytetrational, but the functional equation may be useful ( is not injective). EDIT: just saw your second post  seems you already realized the bit about the functional eq. and how the noninjectivity of contributes. RE: [UFO]  a contradiction in assuming continuous tetration?  Gottfried  08/24/2010 (08/24/2010, 07:19 PM)mike3 Wrote: First off, the graph I give is not regular, it is the Cauchyintegral tetrational (which *may* be equivalent to Kneser's tetrational and the intuitive Abelmatrix one.). This tetrational is real at the real axis and decays to the conjugate fixed points toward . It is, however, asymptotic to regular tetrationals in the imaginary direction (but two different ones, for the two smallestmagnitude conjugate fixed points). Though your graph from the regular looks sort of like mine. (I presume you used the positiveimagpart conjugate fixed point, which would be roughly like the upper half of the Cauchyintegral tetrational.) Hmm  what I see is, that the borders of the plot (the minmax of reals and of imags) differ obviously. Since the x0value 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 minmaxlimits of your graph with that exact value again? I'd like to know, whether the difference between the regular and the cauchyintegralmethod 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 