01/08/2023, 11:11 AM
Mhh I see there is something really deep into this. Many parts still elude me and I can't do nothing. That compact set story and the condition on the domain u sketched are too obscure. I'll get back on the mechanism of \(q_\theta\) and how to reconstruct the original function by, basically, applying the \(q_\theta\) root of the iteration.
Now I see better what you are trying to do... and it is highly related with some work I started to lay out about how rational iterations must be linked together....also the problem of determine that \(q_\theta\) is higly number theoretic... it reminds me of the shit appering in the math behind the Riemann Hypothesis.
Divisibility plays a subtle and fundamental role here... Being able to reframe your discourse in terms of my notes dump on iteration theory thread about iteration will be essential to me. Remember when tommy noticed that it seemed number theory?.
But I need time and calm to do this. Once I do this... which I think is possible and I'll do it for sure in the next days, by setting up a clean and minimal algebraic foundation for this discourse- building up on my algebraic/functorial approach laid out in [ note dump] - i'll be ready to tackle \(\mathbb Q\to \mathbb R\) story of Liouville and so on.
After that only one thing is missing...the meaning of the melling trasform.... recently I watched a quick intro into fourier transforms... using quantum mechanics. The main point i got home was that... you can describe the fundamental objects of QM, instead of by points moving and drawing paths in the space, like in classical dynamics, you can describe them by a wave position function, like a field over the space, or dually using its momentum function (still totally obscure to me). And the fourier transform gets you from one world to the other and back. Now, the position wave function can be se as periodic if we think of the space as in the game "snake", like repeating forever in every direction (like compactifyng the Gauss plane into the Riemann sphere): so a new tool becomes available. We express the position wave function as a sum of "basic elementary waves", a superposition of waves. This is the fourier series. But we express it in exponential form instead of using sines and cosines. The fourier transform can extract from the superposition of waves the single contribution, the weight, of each wave, like you differ-integral is pulling out the coefficients of the auxiliary series... but interpolating it to complex indexes.
It is clear that QM has its fkn role here... and it is months if not years that you hint at it. I believe that the algebraic/functorial undesrtanding of iteration and dynamics admits a similar dual description... and it has to do with the Algebra/Geometry duality. But it is too early for me... I need to review/being introduced to a fuckton of basic material before I can handle this.
Now I see better what you are trying to do... and it is highly related with some work I started to lay out about how rational iterations must be linked together....also the problem of determine that \(q_\theta\) is higly number theoretic... it reminds me of the shit appering in the math behind the Riemann Hypothesis.
Divisibility plays a subtle and fundamental role here... Being able to reframe your discourse in terms of my notes dump on iteration theory thread about iteration will be essential to me. Remember when tommy noticed that it seemed number theory?.
But I need time and calm to do this. Once I do this... which I think is possible and I'll do it for sure in the next days, by setting up a clean and minimal algebraic foundation for this discourse- building up on my algebraic/functorial approach laid out in [ note dump] - i'll be ready to tackle \(\mathbb Q\to \mathbb R\) story of Liouville and so on.
After that only one thing is missing...the meaning of the melling trasform.... recently I watched a quick intro into fourier transforms... using quantum mechanics. The main point i got home was that... you can describe the fundamental objects of QM, instead of by points moving and drawing paths in the space, like in classical dynamics, you can describe them by a wave position function, like a field over the space, or dually using its momentum function (still totally obscure to me). And the fourier transform gets you from one world to the other and back. Now, the position wave function can be se as periodic if we think of the space as in the game "snake", like repeating forever in every direction (like compactifyng the Gauss plane into the Riemann sphere): so a new tool becomes available. We express the position wave function as a sum of "basic elementary waves", a superposition of waves. This is the fourier series. But we express it in exponential form instead of using sines and cosines. The fourier transform can extract from the superposition of waves the single contribution, the weight, of each wave, like you differ-integral is pulling out the coefficients of the auxiliary series... but interpolating it to complex indexes.
It is clear that QM has its fkn role here... and it is months if not years that you hint at it. I believe that the algebraic/functorial undesrtanding of iteration and dynamics admits a similar dual description... and it has to do with the Algebra/Geometry duality. But it is too early for me... I need to review/being introduced to a fuckton of basic material before I can handle this.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)