A function \( G=\tau^{-1}\circ F\circ \tau \) is called a conjugate of the Function \( F \), where \( \tau \) can be an arbitrary other function. With respect to iteration, where we know that \( G^{\circ t}=\tau^{-1}\circ F^{\circ t}\circ \tau \), one always tries to reduce the iteration to the iteration of a simpler function.
For example it was found by Scheinberg [1], that each formal powerseries \( f=\sum_{i=1}^\infty f_i x^i \) is conjugate to \( ax+bx^n+cx^{2n-1} \) for appropriate complex \( a,b,c \) and natural \( n \).
[1] Scheinberg, Power series in one variable, J. Math. Anal. Appl. 31 (1970), 321-333.
For example it was found by Scheinberg [1], that each formal powerseries \( f=\sum_{i=1}^\infty f_i x^i \) is conjugate to \( ax+bx^n+cx^{2n-1} \) for appropriate complex \( a,b,c \) and natural \( n \).
[1] Scheinberg, Power series in one variable, J. Math. Anal. Appl. 31 (1970), 321-333.