On the uniform distribution of orbits of free group and semigroup actions on a plane (Q2783049)

Consider an action of the free group with two generators on the Euclidean plane \(\mathbb R^2\) by orientation-preserving transformations. It is proved that every orbit of the action is either uniformly distributed (on \(\mathbb R^2\)) or nowhere dense. The same holds for the actions of the free semigroup with 2 generators and some special actions of the free semigroup with 4 generators. This extends some results obtained earlier in [\textit{D. A. Kazhdan}, Trans. Mosc. Math. Soc. 14, 325--332 (1965); translation from Tr. Mosk. Mat. Obshch. 14, 299--305 (1965; Zbl 0225.26018); \textit{Y. Guivarc'h}, in Théorie ergodique (Actes Journées Ergodiques, Rennes, 1973/1974, 131--142. Lect. Notes Math. 532, Springer, Berlin (1976; Zbl 0368.28024)].NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].