Theory:

Let \( \{ \zeta_1 , ……., \zeta_m \} \) be a ratinally indifferent cycle for R and let the multiplier of \( R^m \) at each point of the cycle be \( exp \frac {2 \pi i r}{q} \) where \( (r,q) =1 \) . Then \( \exists \ k \in Z \) and\( mkq \) distinct component \( F_1 , F_2 , ….. , F_{mkq} \) s.t. at each \( \zeta_j \) there are exactly \( kq \) of these component containing a petal of angle \( \frac {2 \pi}{kq} \ at \ \zeta \) .

Further R acts as a permutation J on \( F_1 , F_2 , ….. , F_{mkq} \) where J is a composition of k disjoint cycles of length mqJ a petal based at \( \zeta_j \) maps under R to a petal based at \( \zeta_{j+1} \)

Petal theorem :

As there are \( K_j \) such cycles of components for the rationally indifferent cycle \( c_j \) , we see that there are at least \( \sum_{j} \) critical points of P in \( C \) thus \( \sum k_j \leq d-1 \Rightarrow \) we can take the uppper bound to be \( N(d-1) \)