# Problem 39

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Ambrose-Kakutani Theorem for $\mathbb{R}^n$ actions

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See [1].

### References

1. [rudolph1989markov] Rudolph DJ.  1989.  Measure and Measurable Dynamics: Proceedings of a Conference in Honor of Dorothy Maharam Stone, Held September 17-19, 1987. 94:271.

### In [1] and [2], Ambrose and

In [1] and [2], Ambrose and Kakutani showed that every measure-preserving (m.p.) action of $\mathbb{R}$ can be represented as a suspension flow over a  finite m.p. transformation.  Wagh [3] carried out an analogue of the Ambrose-Kakutani theorem for Borel measurable flows on a standard Borel space has a representation as a suspension.  Extensions of the Ambrose-Kakutani and Wagh work to semiflows can be found in [4].  Bowen’s question regards the existence of a similar representation theorem for commuting flows.

In [5] and [6], Rudolph showed that every free, m.p. action of $\mathbb{R}^n$ factors onto an action given by translations on a space of rectangular tilings of $\mathbb{R}^n$.   The base points of the tiles form a cross-section for the action, and one obtains an Ambrose-Kakutani style representation of an $\mathbb{R}^n$-action as a suspension.

Also of note is Rudolph’s improvement of the Ambrose-Kakutani theorem (see [7] and [8]), in which he showed that the height function for the representation of an m.p. flow as a suspension can be chosen that it takes only two values (1 and any irrational alpha > 0).  Versions of Rudolph’s result for $\mathbb{R}^d$ actions can be found in [9].

[1]  W. Ambrose.  Representation of ergodic flows.  Ann. of Math II 42 (1941), 723-729.
[2] W. Ambrose and S. Kakutani.  Structure and continuity of measurable flows.  Duke Math. J. 9 (1942) 25-42.
[3] V. Wagh.  A descriptive version of Ambrose’s representation theorem for flows.  Proc. Indian Acad. Math. Sci. 98 (1988), 101-108.
[4] D. McClendon.  An Ambrose-Kakutani representation theorem for countable-to-1 semiflows.  Disc. Cts. Dyn. Syst. Ser. S 2 (2009), 251-268.
[5] D. Rudolph.  Rectangular tilings of $\mathbb{R}^n$ and free $\mathbb{R}^n$-actions.  In Dynamical Systems (College Park, MD 1986-87), volume 1342 of Lecture Notes in Math., pages 653-688.  Springer, Berlin, 1988.
[6] D. Rudolph.  Markov tilings of $\mathbb{R}^n$ and representations of $\mathbb{R}^n$ actions.  In Measure and measurable dynamics (Rochester, NY 1987), volume 94 of Contemp. Math., pages 271-290.  Amer. Math. Soc., Providence, RI, 1989.
[7] D. Rudolph.  A two-valued step coding for ergodic flows.  Math. Z. 150 (1976), 201-220.
[8] U. Krengel.  On Rudolph’s representation of aperiodic flows.  Ann. Inst. H. Poincare Sect. B. (N.S.) 2 (1976).
[9] B. Kra, A. Quas and A. Sahin.  Rudolph’s two step coding theorem and Alpern’s lemma for $\mathbb{R}^d$ actions.  Trans. Amer. Math. Soc. 367 (2015) 4253-4285.