half iterate of $x^2+c$ I'm looking for literature on fractional iterates of $x^2+c$, where c>0.  For c=0, generating the half iterate is trivial.
$$h(h(x))=x^2$$
$$h(x)=x^{\sqrt{2}}$$
The question is, for $c>0,$ and $x>1$, when is the half iterate of $x^2+c$ smaller than the half iterate of $x^2$?  We know that the full iterate is always larger, since $x^2+c>x^2$, for $c>0$, and $x>1$. Intuitively, one would think that the half iterate of $x^2+c$ would also always be larger, but I believe I have found some counter examples.
In examining the parabolic case for $c=0.25$, I believe $x=800000000$ is a counter example.  $800000000^{\sqrt{2}} \approx 3898258249628$, but I calculate the half iterate of $f(x)=x^2+0.25$, $h_{x^2+0.25}(800000000) \approx 3898248180100$, which is smaller.  
For $c=0$, this is the equation for the superfunction which can be used to calculate fractional iterations.  $f(x)=x^2$, and $g(x) = f^{o x}$, $g(z) = 2^{2^z}$.  For $c=0.25$, this is the parabolic case, which has been studied a great deal in understanding the mandelbrot set, and the superfunction is entire, and I presume there is a uniqueness criteria.  For $c>0.25$, the problem becomes trickier because $x^2+c$ has complex fixed points, and I am also looking for any literature on unique solutions to calculating real valued fractional iterates for $c>0.25$.   
What I am also interested in is the abel function of $x^2$, which is $\text{abel}(z) = \log_2(\log_2(z))$.   I am interested in the abel function of $x^2$ composed with the superfunction of $x^2+c$.
$$\theta(z)=\text{abel}_{x^2}(\text{superfunction}_{x^2+c}(z))-z$$
As real $z$ increases, if $\theta$ converges to a $1$-cyclic function, as opposed to a constant, then there are counter examples like the one I gave, and sometimes the superfunction is growing slower than $2^{2^z}$, and othertimes it is growing faster, with the two function intersecting each other an infinite number of times.  I'm also wondering if $\theta$ converge to an analytic function?  Any relevant links would be appreciated.
- Sheldon
 A: Arguably an off-subject remark:
If only you relented to allow c < 0, there is the celebrated ("chaotic " logistic map) closed form example (p302) of Ernst Schroeder himself (1870); namely, for
$$
h(x)= x^2-2,
$$
it follows directly that for
$$
y=\frac{x\pm \sqrt{x^2-4}}{2}
$$
that is
$$
x=y+y^{-1},
$$
one has
$$
h(x)=y^2+y^{-2}\equiv h_1(x).
$$
Whence, subscripting the iteration index,
$$
h_n(x)= y^{2^n}+ y^{-2^n}.
$$
This, then, specifies the whole iteration group: so your functional square root is just
$$
h_{ 1/2} (x)=y^{\sqrt 2} +y^{-\sqrt 2}. 
$$
Pardon if the point has been made, explicitly, or implicitly, in the outstanding answers above. If not, it might well offer guidance or continuation ideas.
More formally, in E.S.'s language of conjugacy, $\psi(x)=\frac{x\pm \sqrt{x^2-4}}{2}$, $~f(y)=y^2$, $~f_n(y)=y^{2^n}$; so that $h(x)= \psi^{-1} \circ f \circ \psi (x)$, and $$h_n= \psi^{-1} \circ f_n \circ \psi ~.$$
A conjugacy iteration approximation method is available in our 2011 paper:  Approximate solutions of Functional equations.
Apologies if this late lark answer is only proffering  coals to Newcastle, but, in my experience, this is the canonical gambit of chaos discussions--naturally, domains and ranges are chosen suitably for the answer to make sense.
A: a plug 
For some material on fractional iterates of $x^2+c$ see the last section of...
"Fractional Iteration of Series and Transseries" by G. A. Edgar ... LINK
To appear in Trans. Amer. Math. Soc.
A: Similar in nature to Cosmas Zachos' answer using Schroeder's method, it is easy to notice that solutions to
$$f^m(x)=(\underbrace{f\circ\dots\circ f}_m)(x)=x^2+c=\psi(x)$$
all satisfy
$$\psi\circ f=f^{m+1}=f\circ\psi$$
and, assuming non-negativity,
$$f=\psi^{-1}\circ f\circ\psi=\psi^{-n}\circ f\circ\psi^n$$
where $\psi^{-1}(x)=\sqrt{x-c}$. For sufficiently large $x$, one can see that
$$|\psi^{-1}(x+\epsilon)-\psi^{-1}(x)|\le\frac12|\epsilon|$$
and hence as long as $f\circ\psi^n$ is known to be large and $g\circ\psi^n$ is within $\epsilon$ of it for some fixed $\epsilon$ and all large $x$, then the error will vanish as $n\to\infty$, so we may define
$$f=\lim_{n\to\infty}\psi^{-n}\circ g\circ\psi^n$$
Notably, defining $g(x)=|x|^{2^{1/m}}$ and considering $x$ so that $|x|<\psi(x)$ suffices.
For $c>0$, numerical computation tends to suggest this is increasing in $n$ i.e. $g$ is an under-approximation of $f$.
Taking $(m,c,x)=(2,0.25,800000000)$, we get
$$f(x)\simeq3898258249628.005$$
which is equivalent to $g(x)$ for the shown places. A table of values:
\begin{array}{c|c|c}x&g(x)&f(x)\\\hline1&1.00000&1.10352\\10&25.95455&25.99583\\100&673.63884&673.65057\\1000&17483.99554&17483.99862\\10000&453789.29806&453789.29886\end{array}
Try it online!
A: Remark: Shel, possibly I misunderstood something in your post and this pictures here may be completely crap. I expected diff/theta-function-curve crossing the x-axis, but see only the wobbling around a certain y-value. So if this is all wrong, please let me know and I'll improve or delete this post 

An image for the theta-function in your (Sheldon's) original post. I understand the z-parameter in the theta-function as "height"-parameter, when some number $x_0$ is iterated $h$ - (or $z$ -) times to the number $x_h$ .
here is how I implemented the diff-function:  
{shtheta(h,x0=1)= local(a,xh,h1,l2=log(2));
  xh = iterateByAbelfunction(x0,h);
  h1 = log(log(xh)/l2)/l2;  \\ h1 should give the height-difference in terms of 
         \\ the other function $x^2$
  return(h1-h);}

Your example of wobbling was at $x_0=800000000$ - here I begin at $x_0=60$ and show the iterates in steps of 1/10 up to $x_6$ which crosses your 800000000 at height of about $2.3239$ . This is the blue curve in the first plot. The magenta curve is the equivalent, but begins at $x_0=70$ and it should be a left-shift of the blue curve by some small $h$ (just to improve the visualization of the problem):           

The next picture is the detail of bigger "heights" (from $x_1 \approx 3600 $ on) and the magenta-curve shifted to match at the last point at $h=6$ to make the fine sinusoidal form visible.      


[Added]: Hmm, I think now I understand the question and what's going on better now after some more consideration. And I leave the pictures so far, because they are still informative even if not directly to the point.      

My hypothese for now: the "wobbling" which leads to the change of sign in your theta-function is caused by differences or better by a different behave of the functions when derivatives with respect to the height-parameter are considered. Without exact inspection I assume, that the derivatives of all orders of the $x^2$-function with respect to the iteration-height-parameter are always positive but that of the Abel-iteration may be mixed so that the change of the function-value is not "completely smooth".     
I hope I could made this comprehensible so far, perhaps I can do better later ...

[added2]:  I took a closer look at your theta-function and searched for change-signs earlier than your $x_0 = 8e8 $. I found some, for instance $x_0 = 2000 $ and the first 20 iterates in steps of 1/10. Then I scanned 16 areas beginning at $x_0 = 10^{k/2} $ and iterating from $ x_0 $ 20 times by height of 1/10. Each of the latter trajectories make a line in the following plot, also the lines are normalized such that their amplitude is between $ \pm 1 $. Only that lines are drawn which contain at least one sign-change.     



A: remark: this is the comment "g-coefficients" to Will's answer containing the c-program for the Abel-function. 
@Will: Here is the table of coefficients. (Your numbers are originally given as floats(double)) The first significant coefficients-difference is at f8. I don't know which ones are correct and didn't think about the possible nonsignificance due to the n'th iteration of x towards the fixpoints.
                Helms                  Jagy
               -1  log(x)             -1   log(f) 
               -1    n                -1   n     
                1  x^-1                1   1/f
                0   x^0                0    --
              1/2   x^1              1/2   f
             -1/3   x^2             -1/3   f2
            13/36   x^3            13/36   f3
         -113/240   x^4         -113/240   f4
        1187/1800   x^5        1187/1800   f5
         -877/945   x^6         -877/945   f6
      14569/11760   x^7      14569/11760   f7
  -----------------------------------------------------
   -176017/120960   x^8     532963/24192   f8   *** here it begins to differ
  1745717/1360800   x^9  1819157/1360800   f9
    -88217/259875  x^10     -70379/47250  f10

