On measures invariant under diagonalizable actions: the rank-one case and the general low-entropy method
Many important problems in dynamical systems and its interactions with number theory involve classifying measures invariant under the action of a real-diagonalizable group on some locally compact quotient of a real algebraic group. All the results in this spirit have involved some positive entropy hypothesis, and some of the results have required other mixing hypotheses on the measure. Two important steps in the direction of controlling the hypotheses needed are a ``high-entropy'' method, introduced by \textit{M. Einsiedler} and \textit{A. Katok} [Isr. J. Math. 148, 169--238 (2005; Zbl 1097.37017)] (which requires a genuine action of a multi-dimensional group and more than simply positive entropy for a single element) and a ``low-entropy'' method, introduced by \textit{E. Lindenstrauss} [Ann. Math. (2) 163, No. 1, 165--219 (2006; Zbl 1104.22015)] (in which information about the measure is extracted from properties of a single transformation). Because a typical situation involves a completely unknown measure, these two approaches are complementary, and indeed were both brought to bear in the work of the authors and \textit{A. Katok} on Littlewood's problem [Ann. Math. (2) 164, No. 2, 513--560 (2006; Zbl 1109.22004)]. In this paper the low-entropy method is substantially generalized, with the development of rigidity properties for the measurable factors of the action of a single positive entropy element (for an unknown measure) on certain locally homogeneous spaces \(\Lambda\backslash G\). This is used to classify positive entropy measures invariant under a one-parameter group with an extra recurrence condition when \(G\) is a product with a rank-one algebraic group.
scientific article; zbMATH DE number 5290036