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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 II42 (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.

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See [1].

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## 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 II42(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. S2(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.## Add a new comment