Invariant measures for $1$-dimensional discrete dynamical systems The image below was created using the current release of the 
visualization program 3D-XplorMath (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 carefully---we 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?

 A: 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 non-null), 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 one-dimensional 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 one-dimensional analogue of an SRB measure (after Sinai-Ruelle-Bowen).
