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.
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.
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:
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.
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. \[A^{\circlearrowleft \lambda_u}/{\rm core}(\mathbb N{\rm -Act}) \to A{\rm -Act}\]
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;
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. \[A^{\circlearrowleft \lambda_u}/{\rm core}(\mathbb N{\rm -Act}) \to A{\rm -Act}\]
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)