# System dynamic of space euclidean and hyperbolic tilings

Theorem 2.9. (Rudolph [Rud89]) Suppose $X_{T}$ is a finite local complexity (FLC) tiling space. Then $X_{T}$ is compact in the tiling metric d. Moreover, the action $T$ of $R^{d}$ by translation is on $X_{T}$ is continuous. -Probleme : We substitute $R^{d}$ par hyperbolic space $H^{d}$ ?? we can an answer positive?, in particular d=2,

-Can you help me link to the proof of this theorem or the document: ++Daniel J. Rudolph, Rectangular tilings of $R^{n}$ and free $R^{n}$-actions, Dynamical systems (College Park, MD, 1986–87), Springer, Berlin, 1988, pp. 653–688. ++[Rud89] Daniel J. Rudolph, Markov tilings of $R^{n}$ and representations of $R^{n}$ actions, Measure and measurable dynamics (Rochester, NY, 1987), Amer. Math. Soc., Providence, RI, 1989, pp. 271–290. Merci beaucoup.

cf.Symbolic Dynamics and Tilings of $R^{d}$. E. Arthur Robinson, Jr. [page 5] : http://home.gwu.edu/~robinson/Documents/AMS.pdf

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