A new math function inspired from the Julia's fractals theory under construction I working in progress and can creating a new math branch . It is named ef[P](z) where

(ef) mean exponential fractal where P is the polynom of own Julia Set and z<-C a complex number.

Has those properties: ef[P](z)=P(ef[P](z-1)) for whatever complex number z.

This function is good for who is curious how to solving a functional equations like "f(f(z))=z^2-1 what gives f(z)?"

Moment ago I gave some values: ef[z^2](z)=a^(2^z); ef[z^3](z)=a^(3^z); ef[z^2-2](z)=2cos(aTT(2^z));

ef[z^3-3z^2+3](z)=-1+2cos(aTT(3^z)) due cos(2x)=2cos(x)^2-1 and cos(3x)=4cos(x)^3-2cos(x); lim[x->-oo](ef[P](x)) contain all numbers from the border of the Julia set JP(z)={z|P(P...P(z)...)not infinity}.

Other properties are: ef[P(z)](z)+z0=ef[P(z-z0)+z0](z);Q(ef[P(z)](z)=ef(Q(P(~~Q~~(z)))](z) where ~~Q~~ is inverted Q :

~~Q~~(Q(z))=Q(~~Q~~(z))=z, whatever P,Q the complex functions.

The inversion of ef(z) is lof[P](z) with the egalities ef[P](lof[P](z))=lof[P](ef[P](z))=z whatever z,P. I taken the term "lof" that mean logarithm fractal.

Also ef[z](z)=a; ef[z+k](z)=a+kz and ef[kz](z)=a*k^z. Moment ago I founded term "a" like a integrall constant C used and at the differential equations. The function ef[P](z) has complex period : existing v<-R for whick ef[P](z)=ef[P](z+vi) if P is polynom unliniar (kind 2 and above)

More properties of this function coming soon, I still working to find new properties.

If defineing the function efL[P](z)=ef[P](ln(z)) then efL[P](z) is a periodical function like cos(z) and prod[k<-Z](cos(z+kTTi) if P is polynom, due efL[P](z)= P(efL[P](z/e)) has route on the border of the Julia set owned by the polynom P, if z is included in the line whick contains all periods.

Example: If having f[z^2-1](0)=f(2TT)= (1+sqrt(5))/2=z0; f(TT)=-(1+sqrt(5))/2=-z0; f(TT/2)=sqrt(1+z0)=~~P~~(z0); where P(z)=

z^2-1 and ~~P~~ is the inverse; f(TT/4)=sqrt(1+sqrt(1+z0))=~~P~~(~~P~~(z0)); f(2TT/3)=(1-sqrt(5))/2=z1;P(P(z1))=z1; f(TT/3)=sqrt(1+z1)=~~P~~(z1); f(2TT/7)=z2 (*a solution of equation degree 6 whick maybe without solutions with radicals*) from P(P(P(z2)))=z2 etc due f[z^2-1](z)=f(z/2)^2-1 where f[z^2-1](z)=ef[z^2-1](log[2](z))=efL[z^2-1](e^(log[2](z))).

The determining of this f(i) and more formulas coming soon...