I've been reading about products lately and I realized that continuous iteration is an example of a categorical product. But since I'm still learning about category theory, I have some questions and observations I'd like to ask and confirm.
First, the form in which I'm talking about continuous iteration being a product is in the following sense. Similar to a notation found in dynamical systems, iterates of a function \( f : (X \rightarrow X) \) (or equivalently: \( f \in X^X \)) can be denoted \( \Phi^t : X^X \rightarrow X^X \), because the second iterate (for example) of a function from X to X is still a function from X to X. Using this line of thinking, an orbit of the same function can be denoted \( \Phi_x : X^X \rightarrow X^T \), because the time space is different from the phase space, and the orbit of f from 0 (for example) would be a function from T to X. The general iterate of the function (usually written \( \Phi(t, x) = f^{{\circ}t}(x) \)) can then be redefined as an operator of sorts: \( \Phi : X^X \rightarrow X^{T \times X} \).
Secondly, am I thinking of iteration as a functor between endomorphisms? Or is that another way of looking at it? Or is it simply a morphism between objects in the category of sets?
Third, if you put all of the arrows together, along with \( \pi_1 : X^{T \times X} \rightarrow X^T \) and \( \pi_2 : X^{T \times X} \rightarrow X^X \), then you get almost exactly the definition of a categorical product, except with an X exponentiating every set in the commutative diagram. What I wonder is if this is obvious, or if the non-X version being a categorical product does not impliy the X-exponentiated version is a categorical product?
Fourthly, if continuous iteration really is a categorical product, that means that given an orbit map \( \Phi_x \) where \( x \in X \) and an iterate map \( \Phi^t \) where \( t \in T \) , the general iterate \( \Phi \) is uniquely determined. This makes sense intuitively, which makes me think it is so. Also, given the Abel functional equation, it is possible to reconstruct the general iterate \( \exp_b^t(x) \) given only tetration and the super-log, so this makes me think that all functions with a well-defined orbit will also have a unique general iterate \( \Phi \).
Lastly, it is also known that for analytic iteration to exist uniquely, that the function being iterated must have a hyperbolic fixed point. Does this make iteration fail to be a categorical product? Or does it mean that analytic iteration is only a categorical product over the set of functions with hyperbolic fixed points? This part is really confusing me. Any help or clarification would be most appreciated.
Andrew Robbins
First, the form in which I'm talking about continuous iteration being a product is in the following sense. Similar to a notation found in dynamical systems, iterates of a function \( f : (X \rightarrow X) \) (or equivalently: \( f \in X^X \)) can be denoted \( \Phi^t : X^X \rightarrow X^X \), because the second iterate (for example) of a function from X to X is still a function from X to X. Using this line of thinking, an orbit of the same function can be denoted \( \Phi_x : X^X \rightarrow X^T \), because the time space is different from the phase space, and the orbit of f from 0 (for example) would be a function from T to X. The general iterate of the function (usually written \( \Phi(t, x) = f^{{\circ}t}(x) \)) can then be redefined as an operator of sorts: \( \Phi : X^X \rightarrow X^{T \times X} \).
Secondly, am I thinking of iteration as a functor between endomorphisms? Or is that another way of looking at it? Or is it simply a morphism between objects in the category of sets?
Third, if you put all of the arrows together, along with \( \pi_1 : X^{T \times X} \rightarrow X^T \) and \( \pi_2 : X^{T \times X} \rightarrow X^X \), then you get almost exactly the definition of a categorical product, except with an X exponentiating every set in the commutative diagram. What I wonder is if this is obvious, or if the non-X version being a categorical product does not impliy the X-exponentiated version is a categorical product?
Fourthly, if continuous iteration really is a categorical product, that means that given an orbit map \( \Phi_x \) where \( x \in X \) and an iterate map \( \Phi^t \) where \( t \in T \) , the general iterate \( \Phi \) is uniquely determined. This makes sense intuitively, which makes me think it is so. Also, given the Abel functional equation, it is possible to reconstruct the general iterate \( \exp_b^t(x) \) given only tetration and the super-log, so this makes me think that all functions with a well-defined orbit will also have a unique general iterate \( \Phi \).
Lastly, it is also known that for analytic iteration to exist uniquely, that the function being iterated must have a hyperbolic fixed point. Does this make iteration fail to be a categorical product? Or does it mean that analytic iteration is only a categorical product over the set of functions with hyperbolic fixed points? This part is really confusing me. Any help or clarification would be most appreciated.
Andrew Robbins