Linear fuction

Iterates of $T(z)=A+Bz~$ at $~A\!=\!1$, $B\!=\!2~$; $~y=T^n(x)~$ versus $x$ for various $n$

Linear function is function that can be represented in form

$(1) ~ ~ ~ T(z)=A+B z$

where $A$ and $B$ are constants (for example some complex numbers)

Abelfunction and Superfunction

Superfunction $F$ for the linear function $T$ by (1) can be written as follows:

$(2) ~ ~ ~ \displaystyle F(z)= A \frac{1-B^z}{1-B}$

The corresponding Abel function can be expressed as follows:

$(3) ~ ~ ~ \displaystyle G(z)= \log_B\Big(1+\frac{B-1}{A}z \big)$

Iterates

Iterate of the linear function can be expressed through the superfunction and the Abel function in the standard way,

$(4) ~ ~ ~ \displaystyle T^n(z)=F(n+G(z))$

Substitution of the Superfunction $F$ by (2) and the Abel function $G$ by (3) into equation (4) gives aslo linear function

$(5) ~ ~ ~ \displaystyle T^n(z)= \frac{-A+B^n\Big( A+(-1+B) z\Big)}{-1+B}=A \frac {B^n-1}{B-1} +B^n z$

with new parameters $A \frac {B^n-1}{B-1}$ instead of $A$ in (1) and $B^n$ instead of $B$. With this representation, the number $n$ of iteration has no need to be integer. As other holomophic functions, the linear function can be iterated even complex number of times.

Keywords

Abel function, Holomorphic function, Iteration, Superfunction