Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.


The motivation for this question can be found in: <http://www.stat.cmu.edu/~cshalizi/754/2006/notes/solutions-2.pdf> for the logistic map when $x∈[0,1]$. 


My **question** is: 


Is it possible to define the density of the logistic map for $x<0$ where the map convegres to $-∞$?



Is it possible to define the density of the logistic map for $x<0$?
As indicated by Anthony Quas, if you are looking for an invariant probability measure that is supported on the basin of attraction of infinity, then you will need to put a point mass at infinity and nowhere else. (This is clear because a compact set will enter any neighbourhood of infinity after a sufficiently large - finite - number of iterates.)


