We summarize the main ideas in a series of papers ([20], [21], [22], [5]) devoted to the construction of invariant measures and to the long-time behavior of solutions of the periodic Benjamin-Ono equation.
We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.
A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.
A new sufficient condition for the existence of an invariant measure for Markov operators defined on Polish spaces is presented. This criterion is applied to iterated function systems.
For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.