Consider a random walk on the real line: we are given a finite number of homeomorphisms f_1,...,f_n together with the probabilities p_1,...,p_n of their application. Assume that this dynamics is symmetric: together with any f is present its inverse, and they are applied with the same probability. What can be said about such a dynamics?

My talk will be devoted to a joint result with B. Deroin, A. Navas and K. Parwani. Assuming some not too restrictive conditions, we show that almost surely a random trajectory will oscillate between plus and minus infinity. There is no finite stationary measure, but there is an infinite one. There is a random contraction: trajectories of any two initial points almost surely approach each other, the distance being measured in the sense of a compactification of the line (so that any two points both close to plus or minus infinity are counted as close ones). And finally, after changing variables so that the stationary measure becomes the Lebesgue one, one obtains a dynamics with the Dierriennic property: the expectation of image of any point x equals x.