The image below was created using the current release of the
visualization program 3DXplorMath (available
by clicking here. )
It is an image of the Feigenbaum Tree, on
which is superimposed a numerically computed density
function that we believe represents the invariant measure
of the iteration with the current parameter choice. We would
like to document this carefullywe think we have computed
the density correctly, however we have not seen this measure
mentioned elsewhere and do not know any discussion of how to
compute such a measure. (In particular,the obvious Googling
does not turn up anything.) Does anyone know where to
find more about this?



Apologies if I tell you things you already know. The measure you're describing is an absolutely continuous invariant measure for the map at the current parameter value  this is sometimes referred to as an acim or an acip (the latter being if we assume it's a probability measure). One would of course like to start with some existence result that says such a measure actually exists. The statement that there exists a set of parameter values with positive Lebesgue measure for which the corresponding map has an acim is Jakobson's Theorem: see http://www.scholarpedia.org/article/Jakobson_theorem and references therein. As for more explicit computations (such as how big the set of parameters for which an acim exists is, other than just saying it's nonnull), I know that Stefano Luzzatto has done some work in this direction, but cannot reliably say more than that off the top of my head. Addendum: To throw a few more keywords out there, the acip of a onedimensional map $f$ also arises via the thermodynamic formalism as an equilibrium state of the topological pressure for the potential function $\log f'$, and is the onedimensional analogue of an SRB measure (after SinaiRuelleBowen). 

