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Triggered by a question in the math.stackexchange-forum I looked at real fixpoints of b^b^x and found that for the smallest b<e^-e have three real fixpoints (while b^x has only one).

See a short compilation in:
https://math.stackexchange.com/a/2494323/1714

I have not yet found a closed-form expression - for instance by the Lambert W as in the fixpoint-problem of the simple case b^x.

Gottfried
(10/29/2017, 11:14 AM)Gottfried Wrote: [ -> ]Triggered by a question in the math.stackexchange-forum I looked at real fixpoints of b^b^x and found that for the smallest b<e^-e have three real fixpoints (while b^x has only one).

See a short compilation in:
https://math.stackexchange.com/a/2494323/1714

I have not yet found a closed-form expression - for instance by the Lambert W as in the fixpoint-problem of the simple case b^x.

Gottfried

For the two-periodic fixed point pair of  $f(z,k)=-\exp(z)+1+k$
The formal solution for the fixed point of f(x,k) has a power series where the two fixed points are
$L=g(\pm\sqrt{6k});\;\;\;\;f(f(L))=L;\;\;\;\;f(g(\pm\sqrt{6k}),k)=g(\mp\sqrt{6k});$

Then this solutions for the two periodic fixed points L for f(z,k) can be converted to solutions for the two periodic fixed point pair of $a^z$ as follows:

$z=\frac{g(\pm\sqrt{6k})-\ln(-\ln(a))}{\ln(a)};\;\;\;\; k=\ln(-\ln(a))+1;\;\;\;\;a^{a^z}=z$

The definition of the formal series solution for g boils down to this equation; with the x^2/6 term chosen so g x^1 term coefficient=1.   The pari-gp formalperiod2 program returns "1/6" as the x2term.
Also see https://math.stackexchange.com/questions...r-fixed-a0
$-\exp(g(x))+1+\frac{x^2}{6}=g(-x)$

Code:
{g= +x^ 1*  1 +x^ 2* -1/6 +x^ 3*  1/20 +x^ 4* -1/90 +x^ 5*  523/151200 +x^ 6* -23/28350 +x^ 7*  239/1008000 +x^ 8* -19/340200 +x^ 9*  1471949/100590336000 +x^10* -6583/1964655000 +x^11*  94891697/130767436800000 +x^12* -49909/328378050000 +x^13*  18670028801/988601822208000000 +x^14* -520019/241357866750000 +x^15* -88448773393/67224923910144000000 +x^16*  254033333/492370048170000000 +x^17* -15331312862555281/60696039299990814720000000 +x^18*  1418708833351/19449109272763170000000 +x^19* -1799426008377623/80928052399987752960000000 +x^20*  1503421489421/265215126446770500000000 +x^21* -2299035792738061729699/2134218412281997023510528000000000 +x^22*  2825289970112783/13084388263251422617500000000 +x^23*  20135202374763121379/871109556033468172861440000000000 +x^24* -2544602861617837/181168452875788928550000000000 +x^25*  59429596033136172637237727/5070902947582024927861014528000000000000 +x^26* -528038311635637939/143170753680524776956750000000000 +x^27*  13010375215002577166435605861/8823371128792723374478165278720000000000000 +x^28* -38921031170396033459/94664502333562982523803100000000000 +x^29*  100402563034473871496430148044559/873634031969892267532834206929387520000000000000 +x^30* -15748316211640715812312013/550017324845909301985237676542500000000000}

The pari-gp program to calculate the formal g-series for the two periodic fixed points of a generic function gf with a multiplier of (-1) at x=0, is as follows:  Then for example, the error term for fixedaaz(0.04)=0.089600840934760930 is accurate to about about 17-18 decimal digits with a 30 term series.  The other fixed point is fixedaaz(0.04,-1).  For a>exp(-e), the two periodic fixed points are complex conjugate pairs.

Code:
/* formal period2 fixed point for generic function with slope=-1,  2-cyclic solution  */ /* gs=formalperiod2(-exp(x)+1,30)[1]; */ /* returns [series,x2term] */ \ps 31 kf(a)=log(-log(a))-1; fixedaaz(a,n) = {   if (n<>-1, n=1);  return((subst(gs,x,n*sqrt(kf(a)*6))-kf(a)-1)/log(a)); } formalperiod2(gf,n) = {   local(i,gs,savegs,zs,z,zt,x2term,m2,s2,r2);   savegs=0;   gs = x+aecoeff*x^2+O(x^4);   z = subst(gf,x,gs)-subst(gs,x,-x)+acoeff*x^2;   zs = subst(polcoeff(z,3),aecoeff,x);   zt = -polcoeff(zs,0)/polcoeff(zs,1);   savegs = x + zt*x^2;   gs=savegs;   z = subst(gf,x,gs)-subst(gs,x,-x)+acoeff*x^2;   zs = subst(polcoeff(z,2),acoeff,x);   zt = -polcoeff(zs,0)/polcoeff(zs,1);   x2term = zt;   m2 = matrix(2,2);   s2 = matrix(2,1);   forstep (i=3,n,2,     gs = savegs + aocoeff*x^i + aecoeff*x^(i+1)+O(x^(i+3));     z  = subst(gf,x,gs) - subst(gs,x,-x) + x2term*x^2;     zs = subst(polcoeff(z,i+1),aecoeff,x);     zs = subst(zs,aocoeff,0);     m2[1,1] =  polcoeff(zs,1);     zs = subst(polcoeff(z,i+1),aocoeff,x);     zs = subst(zs,aecoeff,0);     m2[1,2] =  polcoeff(zs,1);     s2[1,1] = -polcoeff(zs,0);     zs = subst(polcoeff(z,i+2),aecoeff,x);     zs = subst(zs,aocoeff,0);     m2[2,1] =  polcoeff(zs,1);     zs = subst(polcoeff(z,i+2),aocoeff,x);     zs = subst(zs,aecoeff,0);     m2[2,2] =  polcoeff(zs,1);     s2[2,1] = -polcoeff(zs,0);     r2 = matsolve(m2,s2);     savegs = savegs + r2[2,1]*x^i + r2[1,1]*x^(i+1);   );   return([savegs,x2term]); } gs = formalperiod2(-exp(x)+1,30)[1];