Conjugation of iteration to simpler semigroups.ChatGPT
Conjugation of iteration to simpler semigroups is a general method in the theory of iteration of functions.
The idea is to reduce the iteration of a complicated transfer function \(T\) to the iteration of a simpler family of maps forming a semigroup.
Basic scheme
Let \(T\) be a function acting on some domain. Suppose there exists an invertible function \(G\) such that
\[ G(T(z)) = \Phi(G(z)), \]
where \(\Phi\) is a simpler map whose iterates are explicitly known.
Then \(T\) is said to be conjugate to \(\Phi\), and the iterates of \(T\) satisfy
\[ T^n(z) = G^{-1}(\Phi^n(G(z))). \]
If the semigroup \(\{\Phi^c\}\) is defined for non-integer \(c\), this formula also defines fractional and complex iterates of \(T\).
Classical cases
Several classical functional equations correspond to specific choices of \(\Phi\).
Translation semigroup
If \[ \Phi(w) = w+1, \] then the conjugation equation becomes the Abel equation
\[ G(T(z)) = G(z)+1. \]
Here \(G\) is called the Abel function, and the semigroup is the translation group
\[ w \mapsto w+c. \]
Multiplication semigroup
If \[ \Phi(w) = s w, \] the conjugation equation becomes the Schroeder equation
\[ G(T(z)) = s G(z). \]
The normalized solution is the Koenigs function, and the semigroup is
\[ w \mapsto s^c w. \]
Power semigroup
If \[ \Phi(w) = w^k, \] one obtains the Boettcher equation
\[ G(T(z)) = G(z)^k. \]
This corresponds to iteration near a superattracting fixed point.
Fixed point classification
Let \(z_0\) be a fixed point of \(T\), and let \(s=T'(z_0)\).
Then locally:
- \(s=0\) → conjugation to power map → Boettcher equation.
- \(s=1\) → conjugation to translation → Abel equation.
- \(s\ne 0,1\) → conjugation to multiplication → Schroeder equation.
Thus the type of semigroup is determined by the multiplier at the fixed point.
Semigroup viewpoint
In all cases, the complicated nonlinear iteration of \(T\) is reduced to a simple algebraic semigroup:
- additive semigroup,
- multiplicative semigroup,
- power semigroup.
This unifies the classical functional equations used in regular iteration theory.
Generalizations
More generally, one may consider equations of the form
\[ G(T(z)) = \Phi(G(z)), \]
for an arbitrary function \(\Phi\). In this sense, the classical equations are special cases of conjugation to explicitly solvable semigroups.
Conceptual role in TORI
Within TORI, this framework provides a unified interpretation of:
All of them arise from conjugating iteration to simpler semigroups.