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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 $-∞$?

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    $\begingroup$ Yes. The density is 0 there. $\endgroup$
    – Anthony Quas
    Commented Oct 30, 2014 at 18:28
  • $\begingroup$ @ Anthony Quas: How you get this! $\endgroup$
    – Safwane
    Commented Oct 30, 2014 at 18:29
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    $\begingroup$ What is your definition of "density"? $\endgroup$
    – Gerald Edgar
    Commented Oct 30, 2014 at 19:25

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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.)

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