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 [UFO] - a contradiction in assuming continuous tetration? 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!). « Next Oldest | Next Newest »