Parabolic implosion - from discontinuity to renormalization

Limit dynamics
f 0 ◦ g β = g β ◦ f 0 = g β+1
in K f0
〈f 0 ,g β 〉 = {f0 n ◦ gm β
: (m = 0 and n ≥ 0) or (m >0 and n ∈ Z)}
(when m>0, each element is defined in an open subset of C)
K(〈f 0 ,g β 〉)=C {z : ∃h = f n 0 ◦ gm β ∈〈f 0,g β 〉,h(z) ∈ C K f0 }
J(〈f 0 ,g β 〉) = closure of {repelling fixed points of h = f n 0 ◦ gm β ∈〈f 0,g β 〉}
J(f 0 ) J(〈f 0 ,g β 〉) ⊂ lim inf
n→∞ J(f n)
⊂ lim sup
n→∞
K(〈f 0 ,g β 〉)=J(〈f 0 ,g β 〉) =⇒
K(f n ) ⊂ K(〈f 0 ,g β 〉) K(f 0 )
lim K(f n) = lim J(f n)=J(〈f 0 ,g β 〉)
n→∞ n→∞
∃h = f n 0 ◦ g m β (∈ 〈f 0 ,g β 〉) has an attracting fixed point
=⇒ lim
n→∞ K(f n)=K(〈f 0 ,g β 〉) and
lim J(f n)=J(〈f 0 ,g β 〉)
n→∞