# A Complex Dynamics Problem – Mathematics Stack Exchange

assume $$f (z) = frac {z} {2} + z ^ 2$$ and $$g (z) = frac {z} {2}$$. I want to show that there are some $$r> 0$$ and an analytic function $$phi: mathbb {D} _r (0) to mathbb {C}$$ such as $$(g circ phi) (z) = ( phi circ f) (z)$$ for everyone $$z in mathbb {D} _r (0)$$.

I wrote a series of power $$phi (z) = z + sum_ {n = 2} ^ infty a_nz ^ n$$ and tried to solve for a recursion that the coefficients satisfy. According to the index provided, the term $$a_n$$ will be expressed in terms of $$a_j, ldots, a_ {n-1}$$, or $$j = lceil frac {n} {2} rceil$$. Then, it should be inductively proven that for each $$n$$ we have $$a_n leq16 ^ n$$.

This is a special case of König's famous linearization theorem (1884) derived from a complex dynamics.