Today I read a paper titled “A characterization of the set of fixed points of the Quicksort transformation”
The abstract is:
The limiting distribution \mu of the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T — unique, that is, subject to the constraints of zero mean and finite variance.
We show that a distribution is a fixed point of T if and only if it is the convolution of \mu with a Cauchy distribution of arbitrary center and scale.
In particular, therefore, \mu is the unique fixed point of T having zero mean.