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Invariant measures and long-time behavior for the Benjamin-Ono equation

Yu Deng, Nikolay Tzvetkov, Nicola Visciglia (2014)

Journées Équations aux dérivées partielles

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.

Invariant measures and the compactness of the domain

Marian Jabłoński, Paweł Góra (1998)

Annales Polonici Mathematici

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 V [ 0 , x ] ( 1 / | τ ' | ) 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.

Invariant measures for iterated function systems

Tomasz Szarek (2000)

Annales Polonici Mathematici

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.

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