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Factors of horocycle flows. (English) Zbl 0536.58029

The author classifies up to isomorphism all factors of the classical horocycle flow on the unit tangent bundle of a surface M with constant negative curvature and finite area. The results are in sharp contrast to the situation for the geodesic flow in the same phase space.
A measure preserving flow \(\{S_ t\}\) on a probability space \((Y,\nu)\) is called a factor of a measure preserving flow \(\{T_ t\}\) on a probability space \((X,\mu)\) if there is a measure preserving map \(\psi: X\to Y\) such that \(\psi(T_ tx)=S_ t(\psi x)\) for all \(t\in R\) and almost all \(x\in X\) relative to \(\mu\). The map \(\psi\) is called a conjugacy between T and S, and the flows T and S are called isomorphic if there exists an invertible conjugacy \(\psi\). Conjugacies \(\psi_ 1,\psi_ 2\) between T and S are called equivalent if \(\psi_ 2=B\cdot \psi_ 1\cdot A,\) where A and B are invertible self conjugacies of T and S respectively. \(\pi\) (T,S) will denote the set of these conjugacy equivalence classes. Natural problems for a given flow T on a measure space \((X,\mu)\) are
a) classify up to isomorphism all factors of T and
b) for a given factor S describe the set \(\pi\) (T,S).
The author gives precise answers to both problems when T is the classical horocycle flow.
Let \({\mathcal T}\) denote the set of torsion free lattices in \(G=SL(2,{\mathbb{R}})\); that is, the set of discrete subgroups \(\Gamma\) of G such that the coset space \(\Gamma \backslash G\) is a smooth 3-manifold of finite G-invariant measure. The space \(\Gamma\backslash G\) can be identified with the unit tangent bundle SM of the surface \(M=H/\Gamma\) with finite area and constant curvature -1, where H deotes the hyperbolic plane. The natural Liouville measure \(\mu\) on \(SM=\Gamma \backslash G\) is invariant under both the geodesic flow \(\{g^ t\}\) and the horocycle flow \(\{h^ t\} (break?) given\) respectively by \(g^ t(\Gamma g) = \Gamma g \begin{pmatrix} \exp(t) & 0 \\ 0 & \exp(t)\end{pmatrix}\) and \(h^ t(\Gamma g) = \Gamma g \begin{pmatrix} 1 & 0 \\ t & 1 \end{pmatrix}\). The geodesic flow on \(\Gamma \setminus G\) is Bernoulli and has uncountably many nonisomorphic factors for every lattice \(\Gamma \in {\mathcal T}\). Moreover for any two lattices \(\Gamma_1,\Gamma_2\in {\mathcal T}\) the corresponding geodesic flows are isomorphic and the space of equivalence classes of their conjugacies is uncountable. The horocycle flow behaves much differently as can be seen from Theorems 1 and 2 below.
If \(\Gamma_ 1,\Gamma_ 2\) are lattices in \({\mathcal T}\) with \(\Gamma_ 1\subseteq \Gamma_ 2\), then the horocycle flow on \(\Gamma_ 2\backslash G\) is a factor of the horocycle flow on \(\Gamma_ 1\backslash G\) with respect to the conjugacy \(\psi: \Gamma_ 1g\to \Gamma_ 2g\), \(g\in G.\)
Theorem 1. Any factor of the horocycle flow on \(\Gamma\backslash G\) with \(\Gamma \in {\mathcal T}\) is isomorphic to the horocycle flow on \(\Gamma^*\backslash G\) for a suitable lattice \(\Gamma^*\in {\mathcal T}\) with \(\Gamma \subseteq \Gamma^*\). In particular the set of nonisomorphic factors of the horocycle flow on \(\Gamma\backslash G\) is finite and in one-one correspondence with the set of conjugacy classes of lattices in the finite set \(\alpha(\Gamma)=\{\Gamma^*\in T: \Gamma \subseteq \Gamma^*\}\); an earlier result of the author states that the horocycle flows corresponding to lattices \(\Gamma_ 1,\Gamma_ 2\) in \({\mathcal T}\) are isomorphic if and only if \(\Gamma_ 1,\Gamma_ 2\) are conjugate in G. Now let \(\Gamma_ 1,\Gamma_ 2\) be lattices in \({\mathcal T}\) with \(\Gamma_ 1\subseteq \Gamma_ 2\) and let \(h^{(1)}\), \(h^{(2)}\) denote the horocycle flows on \(\Gamma_ 1\backslash G\), \(\Gamma_ 2\backslash G\). If \(c\in G\) is an element such that \(c\Gamma_ 1c^{-1}\subseteq \Gamma_ 2\) then one may define a conjugacy \(\psi_ c\) between \(h^{(1)}\) and \(h^{(2)}\) by setting \(\psi_ c(\Gamma_ 1g)=\Gamma_ 2cg\) for all \(g\in G.\)
Theorem 2. Every conjugacy between \(h^{(1)}\) on \(\Gamma_ 1\backslash G\) and \(h^{(2)}\) on \(\Gamma_ 2\backslash G\) with \(\Gamma_ 1\subseteq \Gamma_ 2\) is equivalent to a conjugacy \(\psi_ c\), where \(c\in G\) is an element with \(c\Gamma_ 1c^{-1}\subseteq \Gamma_ 2.\) Moreover the set \(\pi(h^{(1)},h^{(2)})\) of equivalence classes of conjugacies is finite (the paper gives a more precise form of the latter assertion). In particular if \(\Gamma\in {\mathcal T}\) is maximal and if S is a factor of the horocycle flow h on \(\Gamma\backslash G\), then S is isomorphic to h and \(\pi\) (h,S) contains one element.
The author concludes by extending these results to the time 1 maps \(h_ 1\) of the horocycle flow h.
Theorem 3. Let S on \((Y,\nu)\) be a factor of \(h_ 1\) on \((\Gamma \backslash G,\mu)\) with \(\Gamma\in {\mathcal T}\), and let \(\psi\) : \(\Gamma\backslash G\to Y\) be a conjugacy between \(h_ 1\) and S. Then there exists a measure preserving flow \(\{S_ t\}\) on \((Y,\nu)\) such that S is the time 1 map \(S_ 1\) of \(\{S_ t\}\) and \(\psi\) is a conjugacy between \(\{h_ t\}\) and \(\{S_ t\}.\)
Corollary 3. If S is a factor of \(h_ 1^{(1)}\) on \(\Gamma_ 1\backslash G\) with \(\Gamma_ 1\in T\), then there exists \(\Gamma_ 2\in {\mathcal T}\) with \(\Gamma_ 1\subseteq \Gamma_ 2\) such that S is isomorphic to \(h_ 1^{(2)}\) on \(\Gamma_ 2\backslash G\). If \(\Gamma_ 1\in {\mathcal T}\) is maximal then every factor of \(h_ 1^{(1)}\) is isomorphic to \(h_ 1^{(1)}\).
Reviewer: P.Eberlein

MSC:

37C10 Dynamics induced by flows and semiflows
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry
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References:

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[2] Rohlin, Mat. Sbornik 67 pp 107– (1949)
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