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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?

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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 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).

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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

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