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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.

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