[note dump] Iterations and Actions MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/23/2022, 02:13 PM (This post was last modified: 07/15/2022, 04:10 PM by MphLee. Edit Reason: numbering ) My time is no longer enough to work out properly the topics and concepts emerging in the latest posts. I also have the impression that I'm not going to polish my notes anytime soon, yet I'd like to not forget every little step I made in understanding some issues in the field of superfunctions. When I write in this forum I try to polish the exposition as much as I can and to include a complete reference to old posts and manuscripts: I can't afford to do this anymore and for this I apologize to the forum members. I also can not afford to study the basic math anymore atm, so what follows could be trivial, full of silly observations, or very wrong. I apologize again but I hope that seeding those notes here could be useful for some active forum members like James, who is always ready to harvest ideas and inspiration and whose work has always been inspirational to me. The plan is to edit the first post adding more notes transcriptions, stand-alone theorems and proofs, observations and questions and prep-notes, so don't expect me to reply under this with long explanations since this post serves mostly as a personal memory tool. INDEX2015, September - The lattice of divisibility and fractional iterates 2021, february 26 - preliminary exploration of "intrinsic iteration" 2022, april 22 - expression of a group(monoid) action using superfunctions 2022, april 28 - more on how to express iterates using superfunctions 2022, May 07 - on the method+UPDATE (NEW) (NEW) 2022, april 22 - expression of a group(monoid) action using superfunctions Let's try to make order in the logical structure of the expression of iterates in terms of a given superfunction. The following is an early attempt to study the additional properties of superfunctions when the domain has a group structure, eg. integers, real and complex numbers. Fix $$(A,+_A)$$ to be an abelian group, the following can be extended to noncommutative monoids. Consider two $$A$$-actions, i.e. two objects of $${\rm Set}^{BA}$$ the category of $$A$$-actions: we take the regular left-translation $$\lambda_a(b):=a+_Ab$$ action over the set $$A$$ and an action $$\alpha_a(y)$$  over the set $$Y$$. A morphism between the two actions, i.e. an element $$f\in {\rm Hom}_{{\rm Set}^{BA}}(A^{\circlearrowleft \lambda},Y^{\circlearrowleft \alpha})$$ is a map $$f:A\to Y$$ that is an $$A$$-equivariant map, something that can be tough as a kind of generalized superfunction, that satisfies $$\forall a,b\in A$$ $$f(a+b)=\alpha_a(f(b))$$ Proposition 1: Let $$(f,g)$$ be any pair of functions s.t. $$f\in {\rm Hom}_{{\rm Set}^{BA}}(A^{\circlearrowleft \lambda},Y^{\circlearrowleft \alpha})$$ and $$g:Y\to A$$ inverts $$f$$ on the right, i.e. $$fg={\rm Id}_Y$$ then the action $$\alpha$$ can be expressed as $$\alpha_a(y)=f(a+g(y))$$ Proof: by def. for every $$a\in A$$ we have $$f\lambda_a=\alpha_af$$. Apply $$g$$ on both sides and obtain the expression. $$\square$$ Let's specialize this to the classical case.  If $$A=\mathbb Z$$ then a $$\mathbb Z$$-action over $$Y$$ is just a choice of a bijection $$\alpha:Y\to Y$$ and $$\alpha_n(y):=(\alpha_1)^{\circ n}(y)$$ Corollary 1 Let $$A=\mathbb Z$$. Let $$(f,g)$$ be any pair of functions s.t. $$f\in {\rm Hom}_{{\rm Set}^{B\mathbb Z}}(\mathbb Z^{\circlearrowleft S},Y^{\circlearrowleft \alpha_1})$$ and $$g:Y\to A$$ inverts $$f$$ on the right, i.e. $$fg={\rm Id}_Y$$ then the action $$\alpha$$ can be expressed as $$\alpha_n(y)=f(n+g(y))$$ Question 1 Going from the proposition to the corollary is matter of restriction, but going from the integer case to the general case is matter of extending iteration from integer time to $$A$$-time. When is it possible? I think I can prove formally that when $$f$$ has not only a right inverse, but also a right inverse, i.e. is bijective, we can prove the closed form of the $$A$$ action. The result is already valid in the real, complex case, and is the folklore expression of non-integer iteration using a superfunction/Shroeder function. Question 2 This is an extension problem, that means that in some sense there could be some "obstructions". Cohomology is described sometimes as a way to measure obstruction. Is it possible to apply cohomology here? How? MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 1,057 Threads: 121 Joined: Dec 2010 04/25/2022, 02:17 AM INFO DUMP! Don't worry about accuracy or perfection, Mphlee! JUST INFO DUMP! I'm excited to see what you have!     MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/26/2022, 01:37 AM (This post was last modified: 07/14/2022, 07:35 PM by MphLee. Edit Reason: FOUND ERROR ) 2022, april 23 - actions, iterations and generalized elements (reboot of old notes) I try to link this to the previous so to make the whole more reasonable. Proposition 1: Fix a monoid $$(A,+_A)$$. Every pair of functions $$f,g$$ s.t. $$f:A\to Y$$ and $$g:Y\to A$$ inverts $$f$$ on the left, i.e. $$gf={\rm id}_A$$ , defines a semigroup (!) action $$\gamma_{f:g}:A\times Y\to Y$$ on the set $$Y$$. Not only it is a semigroup action, it also make $$f$$ into an $$A$$-equivariant map of semigroup actions. The last line means that the naturally defined semigroup action $$\gamma_{f:g}$$ is homomorphic to the natural translation action of $$A$$ on itself via the map $$f$$. Proof: define the binary function $$\gamma_{f:g}:A\times Y\to Y$$ as  $$\gamma_{f:g,a}(y):=f(a+g(y))$$. This map is an action, i.e. $$\gamma_{f:g,a+b}(y)=\gamma_{f:g,a}(\gamma_{f:g,b}(y))$$.  Unwrap the definitions to obtain $$(f\lambda_ag)(f\lambda_bg) =f\lambda_a\lambda_bg=f\lambda_{a+b}g$$. To conclude the proof we have to show that $$f\in {Hom}_{A{\rm Act}}(A^{\circlearrowleft \lambda},Y^{\circlearrowleft \gamma_{f:g}})$$, i.e. $$f(a+b)=\gamma_{f:g,a}(f(b))$$ Unwrap the definitions again $$f\lambda_a=f\lambda_a(gf)=(f\lambda_ag)f$$.    $$\square$$ Corollary the map $$\gamma_{f:g}:A\to End(Y)$$ that takes $$a\in A$$ and produces endomaps $$\gamma_{f:g,a}:Y\to Y$$ is a semigroup homomorphism. It sends $$+_A$$ into composition over $$Y$$ . Observation 1 The map $$\gamma_{f:g}:A\to End(Y)$$ is also a monoid morphism precisely when it sends the element $$0_A$$ to the identity. This happens precisely when $$g$$ is also a right inverse of $$f$$. When this happens $$f$$ must be invertible and $$g=f^{-1}$$. Observation 2 ($$A$$-time) Every monoid $$A$$-action over a set $$Y$$ defines a monoid morphism $$A\to {\rm End}Y$$ (from the "monoid of time" to the "monoid of endofunctions over Y"). We can interpret the latter as a kind of $$A$$-time exponentiation of something that acts on $$Y$$ or an algebraic flow over $$Y$$. In symbols we can express this as a one-to-one correspondence In the special case of the integers $$\mathbb Z$$ we have something peculiar that unlocks a whole kind of heuristic/philosophy around this correspondence. Observation 2 (integer-time) If we set $$A=\mathbb Z$$, the monoid morphism $$\mathbb Z\to A$$, induced by a chosen $$\mathbb Z$$-action, is just the iteration/exponential map $$n\to f^{\circ n}$$ and it is a group homomorphism sending $$+_A$$ to composition of bijections. In this special case, the initiality of the integers as a group (CORRECTION: the initiality used here is of the pointed group $$(\mathbb Z,1)$$ in the cat. of pointed groups), i.e. thanks to the recursion theorem, we have that group morphisms $$\mathbb Z\to {\rm Bij}Y$$ are the same as, in one-to-one correspondence with, bijective maps $$f:Y \to Y$$. Philosophy What can we take from the previous one-to-one correspondences? If $$\mathbb Z$$-actions over $$Y$$ are the same as iteration/discrete flow/exponential map of a bijection, and if choosing a map of that kind is the same as choosing a bijection over $$Y$$ then we can can adopt the following philosophy: Given a group $$A$$ we can see $$A$$-actions over a set $$Y$$ as if they are $$A$$-time iterations of something that we can think as a (generalized) bijection over $$Y$$. In particular we can call that thing a generalized element of the group $${\rm Bij}Y$$ (see Generalized element). Key problem of iteration given an exponential map $$f$$ we recover the base "it is exponential of" just by evaluating it at $$1\in \mathbb R$$. The same for functions and integer iterations. We go from a bijection $$f:Y\to Y$$ to its iteration $${\rm ite}_f:\mathbb Z\to {\rm Bij Y}$$ easily, but also given an integer-time action $$\alpha(n,y)$$ we know it is the iteration of some function, and we can recover it just by evaluating it at one $$\alpha (1,y)$$. The key problem is: someone gives us an $$A$$-action over $$Y$$, or equivalently a group homomorphism $${\rm ite}_f:A\to {\rm Bij Y}$$ how can we recover the function $$f:Y\to Y$$ that we believe it is iteration? Every choice of an element $$u\in A$$ extracts from $$\alpha$$ a different bijection $$\alpha_u$$ over $$Y$$. Proposition 2 Every choice of an element $$u\in A$$ defines a process $$u^*: A{\rm - Act}(Y)\to$$ of restriction of an $$A$$-Action over $$Y$$ to a $$Z$$-action over $$Y$$.  Proof: We define $$u^*f:=f\circ u$$ Consider an $$A$$-Action, write it as a morphism $$f:\mathbb Z\to {\rm Bij Y}$$, precompose it with the iterations of $$u$$, i.e. the map $$u:\mathbb Z\to A;\,\, n\mapsto nu$$ $$(u^*f)^n(y):=f_{u(n)}(y)$$  A composition of morphism is still a morphism hence it induces an action. $$\square$$ Example (Sanity check): Consider a $$(\mathbb C,+)$$-action over the complex numbers, e.g. an exponential map $$\alpha(t,z):=b^t\cdot z$$. Every choice of a (additive) group morphism $$\mathbb Z\to\mathbb C$$, or equivalently every choice of an element $$u\in \mathbb C$$ defines a way to restrict the exponential action to the integers: $$u^*\alpha(n,z):=(b^u)^n\cdot z$$ but in this case we are lucky since among all the maps $$\mathbb Z\to\mathbb C$$ there is a special one: the inclusion of the integers $$1:\mathbb Z\hookrightarrow \mathbb C$$, that is precisely the choice of $$u=1$$. In this case we say conclude by regarding $$\alpha(t,z):=b^t\cdot z$$ as THE (one of the) extension(s) of $$1^*\alpha(n,z):=b^n\cdot z$$, an the latter is the iteration of $$b\cdot z$$. Observation 3: For each way $$u$$ to send $$\mathbb Z$$ to the group $$A$$, we get a machine that restricts $$A$$-actions into $$\mathbb Z$$-actions. In other words, for every morphism $$\mathbb Z\to A$$ we obtain a way to restrict $$A$$-iterations to integer-iterations. This process is functorial. Much more is true. Every group morphism $$A\to B$$ defines a functor that mechanically turns $$B$$-iterations into $$A$$ iterations. Question 3: when we can invert the above construction? In general it is not invertible... there can be two reasons:(loss of information/non-injectivity) many $$A$$-iterations could be restricted to the same integer iteration. For example, there are multiple extension of integer tetration. (non-surjectivity) this obstruction happens only if there exists an integer iteration that cannot be extended to $$A$$-time exponent, i.e. there are integer iterations not in the image of $$u^*$$. What is the best inverse? How is this related to the choice of the best superfunction? Question 4 All of the above discussion adopts a kind  of action-point of view, or exp-like. Because it goes from the time to what is to be iterated. It is a contravariant approach that reminds me of constructions like homotopy and homology. But one should study the dual point-of-view too:. Instead fo the maps $$A\to ...$$ we could study maps $$...\to A$$, i.e. the measure-like point of view, or log-like ($$L(fg)=L(f)+_AL(g)$$). In this approach we look for Abel functions, the various sets inherit the additive structure. Is this second p-o-v related with character theory and with cohomology? MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/28/2022, 03:12 AM (This post was last modified: 04/28/2022, 03:53 AM by MphLee.) 2022, april 28 - more on how to express iterates using superfunctions I'm re-reading the last notes and before I dump more info and old notes I'd like to make clear to myself what those two propositions are. They are making clear what is the relation between non-integer iterations and superfunctions. I have more on this relation but now I think I see this from a new angle and this makes me aware of something I was not understanding. The propositions tell us when we can use a superfunction to find closed form to non-integer iterates. All the previous results were about the relation between abstract superfunction theory and general iteration theory. Call abstract superfunction theory the theory that studies solutions $$\chi:X\to Y$$ to functional equations of the form $$\chi f=g \chi$$. This amounts to the study of integer-iterations; General iteration theory instead is about extending the functions $$n\mapsto f^n$$ from the commutative monoid $$(\mathbb N,+,0)$$ to bigger (commutative) monoids $$(A,+_A,0_A)$$. Objects of study of this theory are functions $$\phi:A\to Y^Y$$ that are solution to the equations $$\phi_{0_A}(y)=y\quad\quad and\quad\quad \phi_{a+_Ab}=\phi_{a}\circ \phi_{b}$$ Key observation 1 Note how all of this is purely algebraic, i.e. it works without caring of convergence and analysis. So whatever works here can only be improved/restricted by adding topology, continuity and differentiability (holomorphy) arguments. Key observation 2 Note how the first theory is equivalent to the theory of $$\mathbb N$$-actions. The second is the theory of $$A$$-actions: the first theory is a special case of the second one. The first theory studies integer iterations as objects. The morphism between integer iterations are $$\mathbb N$$-equivariant maps, aka superfunctions/abel function; The second theory studies $$A$$-actions, or $$A$$-iterations, as objects. The morphisms between $$A$$-actions are $$A$$-equivariant maps, i.e. maps that satisfies a bunch of superfunction equations simultaneously. Warning: from here I change notation: I use $$f$$ for the function to be iterated instead of greek letters. I reserve greek letters for solutions of the superfunction equations. Key problem: is superfunction theory enough, abstractly, to give us genuine non-integer iterates? In other words, how much general iteration theoretic information can we extract from the superfunctions theory of an arbitrary map $$f:Y \to Y$$ Example Let's say, for example, we are given a function $$f:Y\to Y$$ and we want to extend the $$n$$ in expression $$f^{\circ n}:Y\to Y$$ to the quaternions $$\mathbb H$$. Can we use superfunction theory? The naive way to go is: you first find a solution $$\chi:\mathbb H\to Y$$ to the functional equation $$\chi(q+1)=f(\chi(q))$$ where $$q\in\mathbb H$$; then you pray for a map $$\psi:Y\to \mathbb H$$ that somehow inverts the first in some subset; and finally you use all of this to define a map $$F(q,y):=\chi(q+\psi(y))$$ and declare this to be one possible way to extend the iteration of $$f$$. When this is true? The two previous posts try to answer this question. Observation A possible answer is that $$F$$ is really iteration when it is an $$\mathbb H$$-action, i.e. it satisfies the flow equation $$F(p+q,y)=F(p, F(q,y))$$. And this is equivalent to $$F_q:=F(q,-)$$ being a semigroup homomorphism in the $$q$$ argument: $$F_0={\rm id}_Y$$ and $$F_{p+q}=F_p\circ F_q$$. The proposition here gives us a sufficient condition for $$F(q,y)$$ to be a (semigroup) $$\mathbb H$$-iteration:if $$\psi$$ is a right inverse of $$\chi$$; i.e. if $$\chi:\mathbb H\to Y$$ is injective; This will only guarantee that the action behaves additively, but in general $$F(1,y)=f(\chi(\psi(y)))\neq f(y)$$ For this to be true we have to ask that the $$0$$ is sent to the identity by $$F$$, but this is equivalent to $$\psi=\chi^{-1}$$. In general we have the Theorem: Fix a monoid $$A$$ and a special element $$u\in A$$: call it the unit of time in $$A$$. Let $$f :Y\to Y$$ be a function. There is a function $$F_{-}:{\rm Hom}_{\mathbb N{\rm - Act}}(A^{\circlearrowleft\lambda},Y^{\circlearrowleft f})^{\rm iso}\to A{\rm - Act}(Y)$$ Every $$\mathbb N$$-equivariant map $$\chi:A\to Y$$ $$\forall a\in A.\,\chi(u+a)=f(\chi(a))$$ that is a bijection defines an extension $$F_\chi$$ of the iterates of $$f$$ from $$\mathbb N$$ to $$A$$ We have a set cover of the set of superfunctions of $$f$$ $$\bigcup_{\chi \in {\rm Hom}_{\mathbb N{\rm - Act}}(A^{\circlearrowleft\lambda_u},Y^{\circlearrowleft f})}{\rm Hom}_{A{\rm - Act}}(A^{\circlearrowleft\lambda},Y^{\circlearrowleft F_\chi})={\rm Hom}_{\mathbb N{\rm - Act}}(A^{\circlearrowleft\lambda_u},Y^{\circlearrowleft f})$$ Proof: by hypothesis $$\chi$$ is bijective, hence has a right and left inverse. By the proposition 1 (here)  this defines an $$A$$-action over $$Y$$ $$F_\chi(a,y):=\chi(a+\chi^{-1}(y))$$ To prove that $$F$$ extends the $$\mathbb N$$-iteration of $$f$$ we apply the hypothesis that $$\chi(u+q)=f(\chi(q))$$: $$F_\chi(nu,y)=\chi(nu+\chi^{-1}(y))=f^{n}(y)$$ the extension morphism is the map $$\mathbb N\to A:n\mapsto nu$$. For the cover property: by proposition 1 (here) each bijective $$\chi$$ is an $$A$$-equivariant by definition. The containment of the union into the set of superfunction is also given by definition. $$\square$$ Problem: this is not much informative per se. Because this works globally only if $$f(y)$$'s dynamics is isomorphic to the dynamics of the map $$S_A(a):=u+a$$: same orbits, same number of fixed points, . In the case of $$A$$ being the reals or the complex this means that $$f$$ is bijective, and has same orbits, and no periodic-points, as the successor function. I worked alot on how to fix this problem. I guess I found a ways to extract some algebraic information by working on what I call intrinsic iteration of a map: I'll dump notes from 2015 and from 2020/21 very soon. To do: The theorem should be stated category-thoeretically. It seems to me one could produce a functor from the coslice category of the core of N-actions under the addition by $$u$$ to the category of A-actions. $$A^{\circlearrowleft \lambda_u}/{\rm core}(\mathbb N{\rm -Act}) \to A{\rm -Act}$$ MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/28/2022, 03:27 AM 2021, february 26 - preliminary exploration of "intrinsic iteration" To complete the previous discussion I add some old "raw" notes about intrinsic iteration. These are not very polished so use at your own risk. The approach is a bit abstract but is something I'd liked to share because idk If I'll ever have the time to work on it again but I think is important for getting at the rational iterates.   intrinsic iteration.pdf (Size: 432.1 KB / Downloads: 97) MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 1,057 Threads: 121 Joined: Dec 2010 04/28/2022, 07:10 AM (04/28/2022, 03:27 AM)MphLee Wrote: 2021, february 26 - preliminary exploration of "intrinsic iteration" To complete the previous discussion I add some old "raw" notes about intrinsic iteration. These are not very polished so use at your own risk. The approach is a bit abstract but is something I'd liked to share because idk If I'll ever have the time to work on it again but I think is important for getting at the rational iterates. God you have horrible handwriting, I thought I was bad. I showed your handwriting to my friend, and it's worse than his, and all my friends make fun of his chicken scratch. Took me longer but I got through it. Keep posting, I feel this is leading to something fantastic. I just want to cheer on the sidelines, so you know someone is reading all this. Regards MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/28/2022, 08:28 AM (This post was last modified: 04/28/2022, 08:36 AM by MphLee.) (04/28/2022, 07:10 AM)JmsNxn Wrote: God you have horrible handwriting, I thought I was bad. I showed your handwriting to my friend, and it's worse than his, and all my friends make fun of his chicken scratch. Took me longer but I got through it. Hahaha, I'd like to be offended by this but, tbh, I often have hard times reading my own handwriting... it's not writing... is more like some performance art where I use my hand to write ideas directly into my brain and then forget about the paper. xD Anyways I apologize, I'll try to laTex it, for respect: these are notes written hastily using a wacom tablet for pc after an eureka moment using a language that is not my native one. I apologize also for the quality (grammar/typos) because under normal circumstances it takes me 4-5 check to make something I write in Italian acceptable... and it takes me 6-7 checks with grammar tools to make my English text just meh-level... so I'm aware that what I'm doing may be a little too much for the poor reader. But I do mostly just to leave something recorded. For this reason I appreciate even more you kind words James: thanks! You just have to skim through... if only one single lemma or one philosophical observation here will be able to inspire someone I'll know that all of this was meaningful. PS: during 2021 I extended partially all of this to cover, and give algebraic foundation to, your theory of iterated composition (the omega notation)! So all of this can be coherently extended to the case we are composing a sequence of functions. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ JmsNxn Ultimate Fellow Posts: 1,057 Threads: 121 Joined: Dec 2010 04/28/2022, 09:01 AM No need to apologize, MphLee.  I know I have you at an advantage, where you have to speak english, and I don't have to speak italian. You present your ideas very clearly. It's not a problem on my end! It wasn't the english in the notes, it was literally the penmanship, lol. I'm highly anticipating what else you have to say! MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 04/28/2022, 09:17 AM (This post was last modified: 04/28/2022, 09:18 AM by MphLee.) K, thx. If there is any doubt with notation, something big I don't explain because I forgot it is non-trivial, just write some terms, I'll add hyperlinks to the post like in wikipedia, where needed. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ MphLee Long Time Fellow Posts: 367 Threads: 28 Joined: May 2013 05/07/2022, 02:04 AM (This post was last modified: 05/13/2022, 08:29 AM by MphLee. Edit Reason: IMPROVED ) 2022, May 07 - bird's-eye view on algebraic iteration (road to reboot of old notes part 2) IMPOVED/COMPLETED This is the intro to the second part of the notes I was rebooting for this thread but for some reasons I did not include into the 23th April post. The main reason is that I felt it was too abstract and I was going too quickly. That's why I'm going to elaborate a bit on some details of the 28th April's post. Let's do just a brief recap leading us up to speed. Let $$A$$ be a monoid: is is a set of things that you can freely combine $$+_A$$ together (closed), where bracketing doesn't matter (associative) and where there is a thing $$0_A\in A$$ that when combined produces no change in the result (identity element). Def. an $$A$$-Action over a set $$Y$$ is a map $$F:A\times X\to X$$ st for all $$a,b\in A;\, y\in Y$$ $$F(0_A,y)=y$$ $$F(a+_Ab,y)=F(a,F(b,y))$$ Def. an $$A$$-iteration over a set $$Y$$ is a map $$f:A\to {\rm End}(Y)$$ st forall $$a,b\in A$$ $$f_{0_A}={\rm id}_Y$$ $$f_{a+_Ab}=f_a\circ f_b$$ Every $$A$$-action defines an unique $$A$$-iteration and every $$A$$-iteration an unique $$A$$-action. So talking about the former is equivalent to talking about the latter. The moral of the story: is that $$\mathbb N$$-iterations - i.e. the theory of superfunctions - are a special case, a boundary case, of the more general theory of $$A$$-iterations(flows), i.e. dynamics where $$A$$ is an arbitrary monoid of time. ^ In the figure, $$A$$ is meant to be an arbitrary monoid. So the interior of the blob tries to express the vast territory of all the possible iterations using exotic monoids of time, and that connects integer, real and complex iterations. The landscape The crucial observation to understand where we are in the landscape of mathematics is: classical dynamics bounds itself to special monoid of time, $$A=\mathbb R,\mathbb C$$. There is a good reason for this limitation: that's because those are monoids endowed with a continuous (topological) structure and allows us to describe and study differential and analytical properties (flows<->infinitesimal generators<->vector fields). We can say that the theory of $$A$$-actions or equivalently, as we have seen, of $$A$$-iterations is the broad land connecting, bridging the gap between, discrete dynamics and continuous dynamics. It is a general theory of dynamics were we allow the monoid of time to be an arbitrary monoid. ^ In the figure, the monoids are depicted as living inside their box, the "world of monoids" and to each of them there is associated an object of a different nature, that lives in the "world of categories" floating above. The monoids in the figure are just examples. $$A$$ is just a generic monoid living in this world. This world must not be visualized as a linear progression of successive extensions, from smaller monoids to bigger monoids, but rather as a grandiose starry sky. Each monoid is a star, each of different magnitude and a different hue. Some of those monoids happen, as the stars, to be part of a constellation. The relations between monoids links them toghether as the lines connecting the points of a constallation. The monoid of natural numbers is like the pole star because allows you to know the position of all the other stars. While the trivial monoid is the sun, because its light it's not useful to learn something about the other stars, it makes them invisible!!! Terminology: with the term "theory of" I'm rather informal. I don't mean a theory in the model theoretic sense. With a theory of $$A$$-actions I just refer to the complete collection of all the possible $$A$$-actions on every possible set. In addition to that I consider a theory to be the organized collection of all these actions: organized means that I consider all the $$A$$-actions and the network of ways they are in relation with each other. I call this big object $$A{\rm -Act}$$ and this express all the information of what it means for $$A$$ to act on something (a set). The same goes for the theory of $$A$$-iterations: I'll denote that theory with $${\rm Set}^{BA}$$ So, when I say that to each monoid $$A$$ we are associating the theory of $$A$$-actions and that it is equivalent to the theory of $$A$$-iterations... I precisely mean the procedures that take $$A$$ as an input and return these massive objects $$A{\rm -Act}$$ and $${\rm Set}^{BA}$$ as outputs and that in some sense to be made precise $$A{\rm -Act}\simeq{\rm Set}^{BA}.$$ Key observation: these procedures are contravariant functors, preasheaves of categories to be precise. This means many things. The most immediate thing is that the network of relations between monoids gets reflected, reversing all the directions, by the functors: all the relations connecting monoids become relations connecting their theories of iterations!!! Great unification: let's consider the big network containing for each possible monoid it's theory of actions. This monstrous "galaxy" of theories includes the theory of $$\mathbb N$$-iterations, of integer and other discrete dynamics but also rational, real and complex iterations. This landscape but also the $$1$$-iterations of the trivial one-element monoid $$1$$: this is just the theory of sets $$1{\rm -Act}\simeq {\rm Set}^{B1}\simeq {\rm Set}$$, a dynamics where every point is a fixed point, aka the dynamics of the identity! Example 1 Just to have a glimpse of the richness just imagine a universe where the time resets back to the beginning every 5 units of time. In that universe we would like to study $$\mathbb Z /5\mathbb Z$$ iterations. Every iteration of that kind would have inevitably periods of length $$5$$ or fixed points. Example 2 Not only that. That framework is so general that we could study a universe that has as monoid of time the monoid of real numbers under multiplication $$\mathbb R^{\rm mul}:=(\mathbb R,\cdot,1)$$. Then you could ask me: "cmon that is just a silly game, give me an example of an $$\mathbb R^{\rm mul}$$-action that is of any interest!!" I can give an entire class of $$\mathbb R^{\rm mul}$$-actions that is fundamental to classical mathematics: real vector spaces. Sure! to have a vector space you need much more data than an $$\mathbb R^{\rm mul}$$-action structure, but it is true that every $$\mathbb R$$-vector space is an $$\mathbb R^{\rm mul}$$-action. How? An $$\mathbb R^{\rm mul}$$-action over a set $$V$$ produces for each $$\lambda\in \mathbb R^{\rm mul}$$ a "scaling function" $$\mu_\lambda:v\mapsto \lambda v$$ that we regard as an iteration with "multiplicative time". $$(\lambda\kappa)v=\lambda (\kappa v)$$ In this class of examples, that is called linear algebra, an orbit under the iteration is just a linear one-dimensional subspace and the dynamics can be thought as the dynamic of linear motion of a vector from the origin of the vector space. A rosetta stone the last example opens a new world of heuristics and analogies that can guide our intuition and that will guide all this thread. \begin{align} \mathbb N{\bf -iteration} && A{\bf -iteration}&&\mathbb R^{\rm mul}{\bf -action}\\ f^n(y)&&F(a,y) && \lambda v \\ y_0-based\, recursion:\mathbb N\to Y&& y_0-orbit\,map:\, A\to Y && parametrization:\mathbb R\to V\\ n\mapsto f^n(y_0)&& a\mapsto F(a,y_0) && \lambda\mapsto \lambda v\\ orbit\, of \, y:\, \{y,fy,f^2y,...\}\subseteq Y && orbit\, of\, y: \{F(a,y)\}_{a\in A}&&linear\, span\, \langle v\rangle\subseteq V \\ ancestor\,relation && A-reachability&&linear\, dependence \\ y\sqsubseteq_f z\,{\rm iff}\,\exists n.\, f^n(y)=z&&y\sqsubseteq_F z\,{\rm iff}\,\exists a.\, F(a,y)=z && \exists \lambda.\, \lambda v=w \,{\rm or }\,v=\frac{1}{\lambda}w \\ connectedness\,relation && A-connectedness&&linear\, dependence \\ y\frown_f z\,\,{\rm iff}\,\exists n,m.\, f^n(y)=f^m(z)&&y\frown_F z\,{\rm iff}\,\exists a,b.\, F(a,y)=F(b,z) && \exists \lambda,\kappa.\, \lambda v=\kappa w \,{\rm or }\,v=\frac{\kappa}{\lambda}w \\ components\, Y /_{\frown_f}&& components\, Y /_{\frown_f} &&projective\, space \, \mathbb PV \\ ??? && ??? && dimension\\ && && \\ superfunction && A-equivariant \, map && \mathbb R-linear \, map\\ \chi:{(X,g)}\to (Y,f)&& \phi:{(X,G)}\to (Y,F) && M:W\to V\\ \chi(g(x))=f(\chi ( x))&& \phi(G(a,x))=F(a,\phi( x)) && M(\lambda w)=\lambda (M w)\\ \forall n.\,\chi g^n=f^n\chi&& \forall a.\,\phi G_a=F_a\phi && \forall \lambda.\,M \lambda =\lambda M\\ && && \\ && {\bf Eigentheory}&&\\ centralizer\, of\, f && centralizer\,of\,\{F_a\}_{a\in A}&& linear\,operators\,over\, V\\ \alpha f=f\alpha&& \forall a.\,\alpha F_a=F_a\alpha && \forall \lambda.\,\Phi(\lambda v) =\lambda \Phi(v)\\ eigenpoint\,y\, of\, \alpha&& eigenpoint \,y\, of\, \alpha && eigenvector \,v\, of\, \Phi\\ y\,s.t.\, \exists n\in\mathbb N.\alpha(y)=f^n(y)&& y\,s.t.\, \exists a\in A.\alpha(v)=F(a,y) && v\,s.t.\, \exists\lambda\in\mathbb R^{\rm mul}.\Phi(v)=\lambda v\\ eigennumber\,n\,associated\, to\, y&& eigentime\,a\,associated\, to\, y &&eigenvalue\,\lambda\,associated\, to\, v\\ spectrum\,of\,\alpha && spectrum\,of\,\alpha && spectrum\,of\,\Phi \\ \sigma (\alpha)=\{n\,:\,\exists v.\,\alpha(y)=f^n(y)\}&& \sigma(\alpha)=\{a\, eigentime \, of \, \alpha\} && \sigma(\Phi)\\ && && \\ && {\bf Geometry} && \\ ??\simeq\mathbb N^d&& ??\simeq A^d && affine\, spaces\, X\simeq\mathbb R^d \\ ??\subseteq \mathbb N^d&& ??\subseteq A^d && affine\, varieties\, X\subseteq\mathbb R^d \\ ??&& ?? && manifolds \\ ??&& ?? && homology/cohomology\\ \end{align} The last lines of this rosetta stone suggests that, in some way, we are extending geometry itself: from real/complex geometry to the geometries of universes where the time is discrete or more exotic monoid $$A$$. Quote:Man, I don't know what kind of weed you're smoking, but that is insane. i) You think I'm high, I tell you this is just the tip of the iceberg. This is how high we can go:ground) Integer iteration: we are limited to $$\mathbb N,\,\mathbb Z$$; 1) Real/ complex iteration: we extend our number system; 2) We realize that this is not about number systems but about monoids. So we realize that the previous cases are particular instances of a general framework that takes monoids and gives theories of $$A$$-iterations where $$A=\mathbb N, \mathbb R,\mathbb C$$; 3) We realize that all we are doing is in fact considering a monoid $$A$$ as a category with a single object. If we relax our understanding of what we accept as the object representing time to other kind of categories, let's say preorders, we obtain a new extension that contains the theories of iterations as special case; 4) When the object of time is a discrete preorder we obtain a theory of complete recursion and of iterated compositions (omega notation); 5) If we extend the object of time to continuous preorders, or real intervals, we obtain the algebraic version of compositional integrals. 6) If we remove all the limitations on what time can be, we just accept arbitrary "categories of time", we obtain the theory of functor categories. That is part of what can be called $$2$$-category theory. 7) Can we apply this on itself and obtain Goodstein's hyperoperations were ranks are arbitrary? Almost... How high are we atm? We are at point 3). How much THC we are ready to take to get as high as 7)? Quote:One problem I have, and it's dumb, but ii) About this I added a remark under the figure you mention. I warn you to do not place the object $$A{\rm -Act}$$ on the same level of those monoids. So... No, there is not a monoid that comes before, in a strict sense. Remember that every "constellation" in the world of monoids gets projected, reflected like mirror and in a reversed direction, on the world where the theories of $$A$$-iterations live. Just like the following illustration, where I show just one of the infinite constellations and its contravariant reflection: iii) About the topology on these things. That is crucial... I don't know how to canonically extract topological structure from that. I have notes on how to extract divisibility information, i.e. information on the rational iterates... There are more ways btw. In classical analysis the topological information is extracted from two informations: the complete order relation of the real numbers and from its field structure. So you use the metric and from there the norm and use them to study sequences by defining convergence first and continuity. Here maybe we should maybe extract some partial-order information from the monoids and from the iterations and maybe use it in  some way. But that is the analytic way... If we instead follow a more Grothendiek inspiration... maybe I should study the mechanism of sheafification and how to extract directly the lattice of open sets from the monoid or from the set equipped with the action and how all of this varies when modulating the monoid of time. tl,dr: It's too early for me to introduce topology at that general level. I believe I'd better go for Jabotinsky's related business and in how to produce vector fields from $$A$$-actions. 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