Hyperbolic iterated functions computations

PrintDynamic[f_Symbol,z_,n_,Ω_,terms_]:=​​ Module{},​​ Clear[DD];​​DD[f,n,Ω,1]=Derivative[1][f][Ω]^n;​​DD[f,m_,Ω,1]:=DD[f,n,Ω,1]/.nm; ​​ Forderv=2,derv≤terms,derv++,​​ Print
∂
{z,derv}
f[
n-1
f
[z]];​​ ​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z];​​​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt];​​​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt]/.zΩ; ​​​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt]/.zΩ/.0/.Derivative[1][f][Ω]ℒ; ​​ ​​ ​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt]/.zΩ/.0/.Derivative[1][f][Ω]ℒ/.
b_n+c_
ℒ

b+c-1
ℒ
b-1
ℒ
-1
(
bn
ℒ
-
n
ℒ
); ​​ ​​ Print
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt]/.zΩ/.0​​ /.Derivative[1][f][Ω]ℒ/.
b_n+c_
ℒ

b+c-1
ℒ
b-1
ℒ
-1
(
bn
ℒ
-
n
ℒ
)/.ℒDerivative[1][f][Ω]; ​​ ​​ DD[f,n,Ω,derv]=
∂
{z,derv}
f[
n-1
f
[z]]//.Derivative[derv][
-1+n
f
][z]//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt]/.zΩ/.0/.Derivative[1][f][Ω]ℒ/.
b_n+c_
ℒ

b+c-1
ℒ
b-1
ℒ
-1
(
bn
ℒ
-
n
ℒ
)/.ℒDerivative[1][f][Ω];​​ ​​ sderv=ToString[IntegerPart[derv]]; ​​ToExpression["DD[f,m_, Ω, derv]:= DD[f,n, Ω,"<>sderv<>" ] /. n  m"];​​ ;​​​​;
General::spell1:Possible spelling error: new symbol name "sderv is similar to existing symbol derv.
Clear[DD,Dyn,Dynamic];
terms=2;​​DD[f,n,Ω,0]=Ω;​​DD[f,n,Ω,1]=Derivative[1][f][Ω]^n;​​DD[f,m_,Ω,1]:=DD[f,n,Ω,1]/.nm;​​If[Abs[Derivative[1][f][Ω]]==1,​​ cyclotomic[f]:=True,​​ cyclotomic[f]:=False​​];
General::spell1:Possible spelling error: new symbol name "cyclotomic is similar to existing symbol Cyclotomic.
Forderv=2,derv≤terms,derv++,​​ term=
∂
{z,derv}
f[
n-1
f
[z]];​​ term=term//.Derivative[derv][
-1+n
f
][z];​​ term=term//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt];​​ term=term//.zΩ;​​ term=term//.
m_
f
[Ω]Ω;​​ term=term//.Derivative[1][f][Ω]ℒ;​​ Print[term];​​ term=term/.0;​​ DD[f,n,Ω,derv]=term//.
b_n+c_
ℒ

b*(n-1)
∑
i=(b-1)*(n-1)
i
ℒ
;​​ DD[f,n,Ω,derv]=Collect[DD[f,n,Ω,derv],
n
ℒ
];​​ ​​ sderv=ToString[IntegerPart[derv]]; ​​ ToExpression["DD["<>sf<>",m_,"<>sΩ<>", derv]:= DD["​​ <>sf<>","<>sn<>", "<>sΩ<>","<>sderv​​ <>" ] /. "<>sn<>"  m "​​ ];​​;​​ ​​Dyn[f,n,Ω]:=​​Sum[DD[f,n,Ω,cnt]cnt!
cnt
(z-Ω)
,{cnt,0,terms}]/.ℒDerivative[1][f][Ω];
Forderv=2,derv≤terms,derv++,​​term=
∂
{z,derv}
f[
n-1
f
[z]];​​ PrintHoldForm
∂
{z,derv}
f[
n-1
f
[z]],"=",term;​​ term=term//.Derivative[derv][
-1+n
f
][z];​​ Print[term];​​ term=term//.Derivative[vcnt_][
-1+n
f
][z]DD[f,-1+n,Ω,vcnt];​​ Print[term];​​ term=term//.zΩ;​​ Print[term];​​ term=term//.
m_
f
[Ω]Ω;​​ Print[term];​​ term=term//.Derivative[1][f][Ω]ℒ;​​ Print[term];​​ term=term/.0;​​ Print[term];​​ DD[f,n,Ω,derv]=term//.
b_n+c_
ℒ

b*(n-1)
∑
i=(b-1)*(n-1)
i
ℒ
;​​ DD[f,n,Ω,derv]=Collect[DD[f,n,Ω,derv],
n
ℒ
];​​;
∂
{z,derv}
f[
n-1
f
[z]]=2
2
′
(
-1+n
f
)
[z]
+2
-1+n
f
[z]
′′
(
-1+n
f
)
[z]
2
-1+n
f
[z]+2
2
′
(
-1+n
f
)
[z]
-1+2n
2
-2+2n
Ω
+2
-1+n
f
[z]
-1+2n
2
-2+2n
Ω
+2
-1+n
f
[Ω]
2Ω+
-1+2n
2
-2+2n
Ω
ℒ+
-1+2n
2
-2+2n
Ω
-1+2n
2
-2+2n
Ω
cyclotomic[f]
cyclotomic[f]
f
f
PrintDynamic[f,z,n,Ω,3];
2
′
(
-1+n
f
)
[z]
′′
f
[
-1+n
f
[z]]+
′
f
[
-1+n
f
[z]]
′′
(
-1+n
f
)
[z]

′
f
[
-1+n
f
[z]]+
2
′
(
-1+n
f
)
[z]
′′
f
[
-1+n
f
[z]]

′
f
[
-1+n
f
[z]]+
-2+2n
′
f
[Ω]
′′
f
[
-1+n
f
[z]]

′
f
[
-1+n
f
[Ω]]+
-2+2n
′
f
[Ω]
′′
f
[
-1+n
f
[Ω]]
-2+2n
ℒ
′′
f
[
-1+n
f
[Ω]]
(-
n
ℒ
+
2n
ℒ
)
′′
f
[
-1+n
f
[Ω]]
(-1+ℒ)ℒ
(-
n
′
f
[Ω]
+
2n
′
f
[Ω]
)
′′
f
[
-1+n
f
[Ω]]
(-1+
′
f
[Ω])
′
f
[Ω]
3
′
(
-1+n
f
)
[z]
′′
f
[
-1+n
f
[z]]
′′
(
-1+n
f
)
[z]+
3
′
(
-1+n
f
)
[z]
(3)
f
[
-1+n
f
[z]]+
′
f
[
-1+n
f
[z]]
(3)
(
-1+n
f
)
[z]

′
f
[
-1+n
f
[z]]+3
′
(
-1+n
f
)
[z]
′′
f
[
-1+n
f
[z]]
′′
(
-1+n
f
)
[z]+
3
′
(
-1+n
f
)
[z]
(3)
f
[
-1+n
f
[z]]

′
f
[
-1+n
f
[z]]+
3
-2+n
′
f
[Ω]
-
-1+n
′
f
[Ω]
+
2(-1+n)
′
f
[Ω]

′′
f
[
-2+n
f
[Ω]]
′′
f
[
-1+n
f
[z]]
-1+
′
f
[Ω]
+
-3+3n
′
f
[Ω]
(3)
f
[
-1+n
f
[z]]

′
f
[
-1+n
f
[Ω]]+
3
-2+n
′
f
[Ω]
-
-1+n
′
f
[Ω]
+
2(-1+n)
′
f
[Ω]

′′
f
[
-2+n
f
[Ω]]
′′
f
[
-1+n
f
[Ω]]
-1+
′
f
[Ω]
+
-3+3n
′
f
[Ω]
(3)
f
[
-1+n
f
[Ω]]

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