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Dynamical Systems and Number Theory
#1
Check out the article on Dynamical Systems and Number Theory . I don't see a direct connection to tetration, but this is a group of folks that might be interested in our work.
Daniel
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#2
(06/01/2021, 08:54 AM)Daniel Wrote: Check out the article on Dynamical Systems and Number Theory . I don't see a direct connection to tetration, but this is a group of folks that might be interested in our work.
Daniel

Great article, thanks for sharing. Probably, I'd say that we might be interested in their work.
They lost me when they introduced the Jacobian of the polynomial but  that's ok hahah Tongue .
It is remarkable that what I'm doing with superfunctions uses alot of discrete dynamics, heigts and torsion points. A bridge toward number theory (who is bridged itself to differential geometry and shapes like tori and stuff like that) seems interesting.

Just an observation. I believe that the approach of this forum to tetration has always been too narrow. This is also a reason why this forum is not mainstream and is looked over by many outsiders as a cave of crackpotish fanatics. It obviously isn't but, I mean, I'm not insulting ppl here since it is already enough hard and time consuming to explore 1 or 2 points of view for non-professionals like us.

Now that the computational and pure dynamical pov on tetration/hyperoperations was explored it is time now to explore the topic from new points of view: JmsNxn's compositional calculus is one great attempt but more is needed on the topological and algebraic pov. For example more should be invested in lifting up Gottfried and Aldrovandi matrix approach to a full formalism that involves linear operators on inf. dimensional spaces.
Another point that imho needs to be developed is the link with set partitions numbers for the coefficients that Daniel studied. Why? Because I feel, conjecture, it is linked deeply with topological invariants and algebraic topology/number theory....

Also much of the known dynamics/iteration theory results here in the forum should be rephrased in a more unified and modern language (category theory).
It's a grandiose program... maybe the forum needs more marketing...

MathStackExchange account:MphLee

Fundamental Law
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#3
Not so sure about what it would have to do with tetration, but this seems related to periodic points of polynomial maps. I don't have a strong familiarity with Elliptic curves, but it seems we're encoding the periodic points of a polynomial (in complex dynamics), with the group structure of Elliptic curves, and points with finite torsion. Further, this seems much closer to "using complex dynamics to solve problems in elliptic curves" rather than "using elliptic curves to solve problems in complex dynamics." Interesting read though.

As to adopting more mainstream mathematics; I think this is something which has to happen naturally. Mathematicians are slow to adopt new methods. I mean, no one's even translated Kneser's paper from 1950s; and there only exists scattered interest in the problem. But additionally, the solution of these problems is so specialized (for each solution); that it would take some grand unifying thing before people even cared. Until someone relates tetration to something like, fluid dynamics or the Riemann Hypothesis; unfortunately, no one will care. I, however, think this is a good thing. As over saturation of a field means you're constantly competing to beat other people to the chase. I prefer the small little vacuum of research around tetration.

Honestly, what tetration really needs is attention from someone like John Conway (who unfortunately passed last year)--whose whole schtick was this nonsense mathematics that no one ever thought would have any use. I think iteration theory is something we're just beginning to see become mainstream. And by the time it is, people will look at work that was done amongst those here, and say to themselves "well, I guess they beat us to the chase." Kouznetsov's textbook, for instance, is what we should be working towards; collaborating and collating all the work in a packaged one size serves all. I truly believe that Kouznetsov's regular iteration is the first step towards a "black-box" mechanism in iteration theory.

Regards, James
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#4
(06/01/2021, 09:37 PM)JmsNxn Wrote: Honestly, what tetration really needs is attention from someone like John Conway (who unfortunately passed last year)--whose whole schtick was this nonsense mathematics that no one ever thought would have any use.

Agree... maybe in these times John Baez could do the job.
The problem is that I fear the moment one of those sacred monsters will turn their attention to tetration and hyperoperations. Ofc I'll be happy to see the solutions... but it would be sad for me to not have a rich field where I can play around discovering little elementary things like a lil kid. It would take me the fun and the motivation to see an heavyweight mathematician outperform all my efforts of the last 8 years with the first stroke of pen and finding a solution with the second one.

In fact I imagine someone like Milnor or Tao or a von Neumann taking up your compositional calculus, expanding it and solving with it tetration and a billion of other problems in just few months of research (just a few hours for von Neumann), if they only had the need to do it... (I'm not diminishing you ofc, you are lightyears ahead of me. I'm just saying that the best of us would probably get destroyed by the 10yo version of them).

MathStackExchange account:MphLee

Fundamental Law
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#5
(06/01/2021, 10:25 PM)MphLee Wrote:
(06/01/2021, 09:37 PM)JmsNxn Wrote: Honestly, what tetration really needs is attention from someone like John Conway (who unfortunately passed last year)--whose whole schtick was this nonsense mathematics that no one ever thought would have any use.

...I'm just saying that the best of us would probably get destroyed by the 10yo version of them).

LOL! very much agree Sad
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