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 Iterating at fixed points of b^x Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 10/04/2007, 06:53 AM (This post was last modified: 10/04/2007, 11:26 AM by Gottfried.) Maybe I should -at least to restate my view of things- add the following remark. In tetration we do not append exponents to a tower, but bases. So the "partial towers" of an infinite tower are, using a start-value x and a base b (a) x, b^x, b^b^x, ...^ b^b^x and not (b) b, b^b, b^b^... This is crucial, I think. For (a) we get then, for instance for base b=sqrt(2) the two solutions x=2 -> lim h->oo {b,x}^^h =2 x=4 -> lim h->oo {b,x}^^h =4 A supporting argument for this view is also, that if x is already a tower of base b, then the heights are additive... (c.1) {b,x}^^m={b,{b,y}^^n}^^m = {b,y}^^(m+n) (c.2) {b,x}^^m={b,{b,{b,{b,y}^^n}^^n}^^n}^^n = {b,y}^^(4*n) and this is then also coherent with complex fixpoints and real bases as a multisolution problem, even for the limit for infinite heights of towers. This is also, how the matrix-operator-method works, when used for integer-tetration, although, for the finite integer height we may use associativity to change orders of summation and reflect the approach from the opposite direction. It is possibly a bit better expressed in my operators-treatise. ------------------------ Hmm, to avoid confusion, we should possibly talk of "depth" of a powertower instead of "height" to put the mental focus for the problem at the right side Gottfried Gottfried Helms, Kassel bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 10/04/2007, 01:30 PM If we consider $f(z)=b^z$ then $\lim_{n\to\infty}f^{\circ n}(z)$ is either the lower fixed point of $b^z$ or $\infty$ if we start at a non fixed point $z$. This is because all fixed points of $b^z$ are repelling except the lower real fixed point (which though only exists for $b<\eta$). All fixed points can be reached via iterating a branch of the logarithm except the lower real fixed point can be reached by iterating exponentiation. Interestingly the upper real fixed point can be reached by iterating the logarithm even if we start at real $z$ below the lower real fixed point. So for the case $1 either $f^{\circ n}(z)\to \infty$ or $f^{\circ n}(z)\to a$ where $a$ is the lower real fixed point. However I am not sure in the moment for which area A the first case applies. Surely $f^{\circ n}(z)\to\infty$ for each $z>a_2$ where $a_2$ is the upper fixed point and $f^{\circ n}(z)\to a$ for all other real $z$. But for example $f^{\circ n}(5+i)\to 2$, but $f^{\circ n}(5+0.1i)\to\infty$ for $b=\sqrt{2}$. Jay? Also we didnt consider yet the case of non-real $b$. bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 10/04/2007, 05:54 PM bo198214 Wrote:However I am not sure in the moment for which area A the first case applies. Surely $f^{\circ n}(z)\to\infty$ for each $z>a_2$ where $a_2$ is the upper fixed point and $f^{\circ n}(z)\to a$ for all other real $z$. So then I simply made a fractal of it: The more green the color is the more iterations $n$ does it take that $|f^{\circ n}(z)|>T$ (i.e. the slower it converges to $\infty$) and if it needs more than 10 iterations (i.e. probably it does not converge to infinity but to the lower fixed point) then the color is black. I chose $b=\sqrt{2}$, the presented rectangle is $[4\dots 12] \times [-3\dots 3]$, I made it with FractalExplorer. $T=170$     $T=500$     $T=1000000000$     We see that the convergence to infinity is really chaotic. Gottfried Ultimate Fellow Posts: 767 Threads: 119 Joined: Aug 2007 10/04/2007, 11:05 PM Gottfried Wrote:For (a) we get then, for instance for base b=sqrt(2) the two solutions x=2 -> lim h->oo {b,x}^^h =2 x=4 -> lim h->oo {b,x}^^h =4 Hmm, Andrew, I just see at your site, that this contradicts to one of your identities. You say , at "integer values" (translated to ascii-notation) {b,a}^^oo = b^^oo to which my above definition is then in contradiction. On the other hand, saying "tetration means: appending bases" is also more consistent with the right-to-left-evaluation rule... Hmm. Gottfried Gottfried Helms, Kassel GFR Member Posts: 174 Threads: 4 Joined: Aug 2007 01/30/2008, 10:41 PM (This post was last modified: 01/31/2008, 10:01 AM by GFR.) Concerning: bo198214 Wrote:There is also a mysterious base where the primary fixed point is exactly $i$. This is the case for $i=b^i$ which is clearly satisfied for $b=e^{\frac{\pi}{2}}\approx 4.8104$.Some comments: (1) - General observations The study of the fixpoints of y =f(x) [... simply ...!] means the analysis of the points where f(x) = x, i.e. the intersections of y = f(x) with y = x, the principal diagonal of the xy plane. Fixpoint analysis is an excellent tool used by modern mathematicians for finding serial developments and, particularly, by specialists in analytic functions, recursivists, iterationists, fractalists, as well as experts in AI and NKS matters. The application of this tool to our subject implies the study of the intersections of y = b^x with y = x, for various b, i.e. for finding the values of x for which b^x = x. At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b. (2) - Some calculations attempts Some months ago, a friend of mine asked me, for fun, to estimate the numerical value of 7#3 = 7-tetra-3 = 7^(7^7). My pocket calculatort immediately gave me 7^823543 and, after that, it went to overflow. Then I decided to use Mathematica and I got: 7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages. Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got: rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision. Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) ! Why (... the Hell!), at the same time, equation rho^x = x gives as solution x = {-i, +i} ? Can an infinite tetration of base "rho" softly collapse to a conjugate imaginary unit ? This, of course, while in expression such as y = rho^n superexponent n "goes" to infinity? (2) - Kritik of the Mathematical Reason It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!). Then, our position could be: - to say that they don't; - to say that, unfortunately, they do (reach non-contradictory results). The fact is that, if they do bring us to coherent results, the infinite tetration of base rho (sorry, this is a nightmare of mine) should give: rho#oo = {-i, +i, +oo}. As a matter of fact, in the domain of b > e^(1/e), the "real" solution +oo is (at least, thoretically) always obtainable by an infinite number of iterative calculations and complex solutions are calculable via the inversion of b = y^(1/y), i.e.: y = b^y. The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite? I can't believe it. That would mean a "mathematical fiction" relation of ... order between the real and the complex numbers. Now, Henryk, you are authorized to say: "No, No, No!". Perhaps, I am joking! In conclusion, what shall we do? Forget the relation of order. Please read the attached pdf notes and tell me, if you wish, your reactions. They also contain the plots provided by Gottfried and Andydude. Or, please destroy them ... before reading, if you prefer so. Forgive me for any mistake, omission or lack of precision and remember: "Also Human Ignorance (I mean mine) is a Gift of God!". (Otto von Bismarck, I presume). Thanks! GFR Attached Files   Mysteries.pdf (Size: 136.86 KB / Downloads: 348) bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 01/31/2008, 01:43 PM GFR Wrote:Then I decided to use Mathematica and I got: 7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages. Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got: rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision. Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) !Its really fun to read your post. Quote:At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b. ... (2) - Kritik of the Mathematical Reason It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!). But thats not a critique of *mathematical* reasoning but a critique of *human* reasoning! Mathematical reason goes like that: Let $f$ be a real (or complex) continuous function and suppose that $y=\lim_{n\to\infty} f^{\circ n}(x_0)$ exists (and this does not include infinity) then $f(y)=y$. Proof: Define the sequence $x_n:=f^{\circ n}(x_0)$. Then by assumption $\lim_{n\to\infty} x_n = y$ but we also have by assumption that $\lim_{n\to\infty} f(x_n)= y$. And as $f$ is continuous $y=\lim_{n\to\infty} f(x_n)=f\left(\lim_{n\to\infty} x_n\right)=f(y)$. q.e.d. Also the opposite implication, that from $f(y)=y$ would follow $\lim_{n\to\infty} f^{\circ n}(x_0)=y$, is not true! Thats just the human temptation to always revert implications. Quote:The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite? Yeah the good old infinity. Unfortunately you can not just simply add infinity to the real or complex numbers (it leads to contradictions). They only thing I can add are two references (from my other forum) to similar ideas which however are rather in an undeveloped state: Idea stub 1 which evolved from this thread. (There is especially for Ivars a thread discussing hyperreals on this forum.) And idea stub 2. Quote:Please read the attached pdf notes and tell me, if you wish, your reactions. ... Or, please destroy them ... before reading, if you prefer so. Dunno whether it was a destruction, but I indeed didnt read your attachment yet, but will do it now GFR Member Posts: 174 Threads: 4 Joined: Aug 2007 01/31/2008, 03:07 PM Hi! You see, with the tetrational number notaion, base 2 or base e, respectively, we shall have: 7#3 = 3.75982352678353 x 10^695974 = 1.09635839450984 * 2#5 = 2.65935709200036 * e#3. The number before the # sign is the "tetrational significance" (the significant figures of the tetra-notation), the last exponent of the unhomogeneous tower, or the "tower extension". The first "floor" of the push-down tower, according to Gottfried. GFR GFR Member Posts: 174 Threads: 4 Joined: Aug 2007 01/31/2008, 06:20 PM Hi, Henryk! You mentioned the following "Idea stub 2", concerning a possible extension of the reals to infinite ... entities and, particularly, the use of tetration for clearly representing the "limit epsilon-1" infinite Cantor's ordinal, as follows: bo198214 Wrote:And idea stub 2. I agree that, in general, hyperoperations could be an excellent tool for further "extending epsilon1" , via pentation hexation etc, up to "omegation". We (KAR & GFR) already presented this idea to the NKS Forum (see: Hyperoperations, second progress report). A new "limit" would then appear: w[w]w = omega-omega-omega which could probably be taken for (Capital) OMEGA, the (first?) uncountable infinite ordinal. Good grief. GFR Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/28/2008, 07:37 AM bo198214 Wrote:Here is a zoom of the primary fixed point showing its trajectory for $b=\eta+10\dots \eta+0.05$...Henryk: 1. I need your fixed points. Could you please, describe them and publish somewhere, at least a preprint, in order than I can cite you? I agree fo cite your paper "in preparation", if you post somewhere the draft and promise to submit it soon. 2. I have not yet cleaned up the rest of figures; I see, I do it too slowly. If you agree to work with drafts, download the generators at url=http://www.ils.uec.ac.jp/~dima/TETRATION/LOG/analuxp.tar.gz. « Next Oldest | Next Newest »

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