# 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
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).
I went to a talk by Luzzatto on that topic, and as I recall his rigorous lower bound for the size of the parameter region for which an ACIM exists was something like $10^{-100}$. I do recall that for parameters lying in the region $[1-\varepsilon,1]$, the proportion of parameters which yield an ACIM tends to $1$ as $\varepsilon \to 0$. –  Ian Morris Sep 13 '10 at 9:14