This is meant to be a quick question to begin to rectify my pov, before I begin to study seriously what I need to study (Milnor and diff. geometry in general).
Recently some remarks about non uniqueness were made. I had not the time to understand it properly but I have an image in my mind... something about the direction I should take once I make clear to myself the algebraic approach to iteration/superfunctions.
Algebraic approach. this is very minimal and well behaved. An iteration is an homomoprhism \(f_{t+s}=f_s\circ f_t\), while a generalized superfunction (an \(A\)-equivariant map) is a solution of the equation \(g_s\circ \chi=\chi \circ f_s \).
The problem about this approach is that often such solutions do not exists globally. For this reason I see how we can introduce something "new" like with atlases of manifolds. The question is: is this new? Is this exactly what you are all doing here? Adopting this pov can bring progress?
Definition: let \(f:X\to X\) be a function. Let \(S:A \to A\) the topological domain of time. Define a superfunction atlas of \(f\) to be
an open cover \(U_i\subseteq A\) of \(A\) such each \(S(U_i)\subseteq U_i\) and a family of superfunctions \[\Psi_i:U_i\to X\]
s.t. \[\Psi_i(S|_{U_i}(a))=f(\Psi_i(a))\]
Note that here it is possible that \(\Psi_i:U_i\to X\) and \(\Psi_j:U_j\to X\) may disagree on \(U_{ij}=U_i\cap U_j\). In that case the atlas produces different iterations depending on the element of the cover chosen... but globally we are covering all of \(A\).
We may also ask that some coherence conditions like in atlases of coordinate charts.
Example: We may ask that the restrictions \(\Psi_j |_{U_ij}:U_{ij} \to X\) and \(\Psi_i |_{U_ij}:U_{ij} \to X\) have nice properties, like glueing maps.
Tbh we could also turn this definition upside down asking instead for a cover of \((X,f)\)... but I'm not sure how to go because It is not clear to me what goals we would like to fulfill and what are the obstacles in the classical approach to locally/globally iterating stuff.
ps: I feel that untangling this is the first step before bringing all the algebraic theory to a new level merging topological and analytical phenomena into a new abstract framework.
Recently some remarks about non uniqueness were made. I had not the time to understand it properly but I have an image in my mind... something about the direction I should take once I make clear to myself the algebraic approach to iteration/superfunctions.
Algebraic approach. this is very minimal and well behaved. An iteration is an homomoprhism \(f_{t+s}=f_s\circ f_t\), while a generalized superfunction (an \(A\)-equivariant map) is a solution of the equation \(g_s\circ \chi=\chi \circ f_s \).
The problem about this approach is that often such solutions do not exists globally. For this reason I see how we can introduce something "new" like with atlases of manifolds. The question is: is this new? Is this exactly what you are all doing here? Adopting this pov can bring progress?
Definition: let \(f:X\to X\) be a function. Let \(S:A \to A\) the topological domain of time. Define a superfunction atlas of \(f\) to be
an open cover \(U_i\subseteq A\) of \(A\) such each \(S(U_i)\subseteq U_i\) and a family of superfunctions \[\Psi_i:U_i\to X\]
s.t. \[\Psi_i(S|_{U_i}(a))=f(\Psi_i(a))\]
Note that here it is possible that \(\Psi_i:U_i\to X\) and \(\Psi_j:U_j\to X\) may disagree on \(U_{ij}=U_i\cap U_j\). In that case the atlas produces different iterations depending on the element of the cover chosen... but globally we are covering all of \(A\).
We may also ask that some coherence conditions like in atlases of coordinate charts.
Example: We may ask that the restrictions \(\Psi_j |_{U_ij}:U_{ij} \to X\) and \(\Psi_i |_{U_ij}:U_{ij} \to X\) have nice properties, like glueing maps.
Tbh we could also turn this definition upside down asking instead for a cover of \((X,f)\)... but I'm not sure how to go because It is not clear to me what goals we would like to fulfill and what are the obstacles in the classical approach to locally/globally iterating stuff.
ps: I feel that untangling this is the first step before bringing all the algebraic theory to a new level merging topological and analytical phenomena into a new abstract framework.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)