Geodesic
On the Uniform Distribution of Orbits of Free Group and Semigroup Actions on a~Plane
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 64-95

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The orbits of actions by orientation-preserving motions on the Euclidean plane are considered for the following groups and semigroups: a free group and free semigroup with two generators, and a free semigroup with four generators (only the actions of special type are considered for the latter case). It is established that any orbit is either uniformly distributed over the plane or nowhere dense. 
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     author = {Ya. B. Vorobets},
     title = {On the {Uniform} {Distribution} of {Orbits} of {Free} {Group} and {Semigroup} {Actions} on {a~Plane}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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Ya. B. Vorobets. On the Uniform Distribution of Orbits of Free Group and Semigroup Actions on a~Plane. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Dynamical systems, automata, and infinite groups, Tome 231 (2000), pp. 64-95. http://geodesic.mathdoc.fr/item/TM_2000_231_a2/