Rudolph’s two step coding theorem and alpern’s lemma for ℝdactions

Bryna Kra, Anthony Quas, Ayşe Şahin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Rudolph showed that the orbits of any measurable, measure preserving Rd action can be measurably tiled by 2drectangles and asked if this number of tiles is optimal for d > 1. In this paper, using a tiling of ℝdby notched cubes, we show that d + 1 tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of ℝ2by two rectangles, we show that while for ℝ2actions with completely positive entropy this bound is optimal, there exist mixing ℝ2actions whose orbits can be tiled by 2 tiles.

Original languageEnglish (US)
Pages (from-to)4253-4285
Number of pages33
JournalTransactions of the American Mathematical Society
Issue number6
StatePublished - 2015

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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