10/15/2022, 01:17 AM
Excuse me, I'm not lazy or what... only very busy atm... but I'd like to begin to chew this material... everyone here seems to understand terms and definitions but I'm a total beginner.
What the hell is, formally, a flower? Looking at the pictures it seems the image of a path. In particular it seems it has to do with images of the family of paths \[\gamma_n:\mathbb S^1\to \mathbb C\]
defined as \(\gamma_n:=f^n\circ \gamma_0\) where \(\gamma_0\) is the is the inclusion of the circle as the unit circle, or a scaled version of it.
Question 1: what this family of loops/paths has to do with the formal definition of flowers and petals?
Question 2: in any case, from the algebraic topology pov, each of these \(\gamma_n\) can be sent to their homotopy-class \([\gamma_n]\in \pi_1(\mathbb C)\). In the case of the complex plane, has the behavior of the induced discrete action on the fundamental group of the complex plane being studied? In general, if \(X\) is a topological space equipped with a continuous map \(f:X\to X\), and \(p\in {\rm fix}(f)\), we can define by post-composition an endomap \[{\bar f}:\pi_1(X;p)\to\pi_1(X;p)\] (for a proof use functoriality the first-homotopy group construction).
The question is, what the induced dynamics on the homotopy groups tell us about the original dynamics?
What the hell is, formally, a flower? Looking at the pictures it seems the image of a path. In particular it seems it has to do with images of the family of paths \[\gamma_n:\mathbb S^1\to \mathbb C\]
defined as \(\gamma_n:=f^n\circ \gamma_0\) where \(\gamma_0\) is the is the inclusion of the circle as the unit circle, or a scaled version of it.
Question 1: what this family of loops/paths has to do with the formal definition of flowers and petals?
Question 2: in any case, from the algebraic topology pov, each of these \(\gamma_n\) can be sent to their homotopy-class \([\gamma_n]\in \pi_1(\mathbb C)\). In the case of the complex plane, has the behavior of the induced discrete action on the fundamental group of the complex plane being studied? In general, if \(X\) is a topological space equipped with a continuous map \(f:X\to X\), and \(p\in {\rm fix}(f)\), we can define by post-composition an endomap \[{\bar f}:\pi_1(X;p)\to\pi_1(X;p)\] (for a proof use functoriality the first-homotopy group construction).
The question is, what the induced dynamics on the homotopy groups tell us about the original dynamics?
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)