Rudolph’s Two Step Coding Theorem and Alpern’s Lemma for R Actions
نویسندگان
چکیده
Rudolph showed that the orbits of any measurable, measure preserving R action can be measurably tiled by 2 rectangles and asked if this number of tiles is optimal for d > 1. In this paper, using a tiling of R by notched cubes, we show that d + 1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of R by two rectangles, we show that while for R actions with completely positive entropy this bound is optimal there exist mixing R actions whose orbits can be tiled by 2 tiles.
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