Hy Tommy,
you are right. To be precise, it's not totally number theory but just the part about divisibility. Why this kind of math appears when we study iteration?
I'd explain it as follows.
Divisibility is basically the math of ideals in rings. Ideals are mutiplicative objects, i.e. they relates to the multiplicative structure of a ring. Primality, irriducibility and stuff like that are ideal theoretic in nature.
The passage is a bit tricky: integer itaration is about the action of the monoid of positive integers, fractional iteration is a bout actions of the abelian group of rational numbers (under addition), continuous iteration (aka dynamics) is about continuous actions of the additive ab. grp. of real numbers... and so on.
In general we could say that dynamics/iteration is about the action of a monoid over something: the monoid is the place where the time lives. But to be less exotic we can restrict ourselves to commutative and reversible time (abelian and group). Now the point is, when the monoid is an abelian group, its endomorphisms form a non commutative ring where addition is defined pointwise from the addition of the group, and the multiplication is the composition (endomorphism=distributivity). Eg. the ring of endomorphisms of the abeliang group integers under addittion is essentially the ring of integer numbers (here is where number theory kicks in). \[{\rm End}(\mathbb Z,+)\simeq (\mathbb Z, +,\cdot)\]
Another tricky passage is needed to have the full picture. The multiplication of the ring of endomorphism of the abelian group of time \(A\)... acts on "\(A\)-timed" iterations by something that in, the special case of integer iteration, is the power map \(f\mapsto f^{\circ k}\). In other words, the complete information of the \(A\)-iteration of something is contained in the ring of endomorphisms of \(A\). The divisibility relation in the ring is reflected somehow in the root relation between the iterates.
Notice Tommy, that the purpose of this serie of posts is just writing down some of my old notes I want to share, some of these are just incomplete ideas.
The value of this is subjective I guess. I like to save some of these ideas so that maybe I can extract something valuable later, when I have more time. Also, I find usefull to unify known algebraic facts about iteration using an unique language.
Regards.
ps. I Just hope we are not at this point already
![[Image: 13615158-160776897675046-4728901722514233204-n.jpg]](https://i.ibb.co/kx9030g/13615158-160776897675046-4728901722514233204-n.jpg)
Quote:This looks like number theory
you are right. To be precise, it's not totally number theory but just the part about divisibility. Why this kind of math appears when we study iteration?
I'd explain it as follows.
Divisibility is basically the math of ideals in rings. Ideals are mutiplicative objects, i.e. they relates to the multiplicative structure of a ring. Primality, irriducibility and stuff like that are ideal theoretic in nature.
The passage is a bit tricky: integer itaration is about the action of the monoid of positive integers, fractional iteration is a bout actions of the abelian group of rational numbers (under addition), continuous iteration (aka dynamics) is about continuous actions of the additive ab. grp. of real numbers... and so on.
In general we could say that dynamics/iteration is about the action of a monoid over something: the monoid is the place where the time lives. But to be less exotic we can restrict ourselves to commutative and reversible time (abelian and group). Now the point is, when the monoid is an abelian group, its endomorphisms form a non commutative ring where addition is defined pointwise from the addition of the group, and the multiplication is the composition (endomorphism=distributivity). Eg. the ring of endomorphisms of the abeliang group integers under addittion is essentially the ring of integer numbers (here is where number theory kicks in). \[{\rm End}(\mathbb Z,+)\simeq (\mathbb Z, +,\cdot)\]
Another tricky passage is needed to have the full picture. The multiplication of the ring of endomorphism of the abelian group of time \(A\)... acts on "\(A\)-timed" iterations by something that in, the special case of integer iteration, is the power map \(f\mapsto f^{\circ k}\). In other words, the complete information of the \(A\)-iteration of something is contained in the ring of endomorphisms of \(A\). The divisibility relation in the ring is reflected somehow in the root relation between the iterates.
Notice Tommy, that the purpose of this serie of posts is just writing down some of my old notes I want to share, some of these are just incomplete ideas.
The value of this is subjective I guess. I like to save some of these ideas so that maybe I can extract something valuable later, when I have more time. Also, I find usefull to unify known algebraic facts about iteration using an unique language.
Regards.
ps. I Just hope we are not at this point already
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)