# zbMATH — the first resource for mathematics

Geodesic laminations as geometric realizations of Pisot substitutions. (English) Zbl 0963.37013
Consider the Pisot substitution $$\pi$$ on $$\{1,\dots, n\}^\mathbb{N}$$ defined by substituting, for $$2\leq j\leq n$$, $$1j$$ to each $$(j-1)$$, and 1 to each $$n$$. Let $$\Omega$$ be the closure of the orbit under the shift map $$\sigma$$ of the unique fixed point of $$\pi$$. By means of an interval exchange map on the circle, the author extends $$\Omega$$ to $$\Omega'\subset \{1,\dots, n\}^\mathbb{Z}$$, on which $$\sigma$$ still operates, and he constructs a closed self-similar set $$\Lambda$$ of disjoint geodesics in the hyperbolic disk, a Hausdorff-like measure $$\mu$$ on $$\Lambda$$, and a continuous $$\mu$$-preserving transform $$F$$ on $$\Lambda$$, such that the dynamical systems $$(\Lambda,F)$$ and $$(\Omega',\sigma)$$ are topologically conjugate.

##### MSC:
 37B10 Symbolic dynamics 37E10 Dynamical systems involving maps of the circle 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Full Text: