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**Horocycle flows, joinings and rigidity of products.**
*(English)*
Zbl 0556.28020

This is a fairly long and technical paper in which a number of interesting results are proved. The paper is divided into two parts: part I deals with joinings of horocylce flows; part II deals with rigidity of products of horocycle flows. We quote a sample result from each part which, hopefully, will convey the flavour of the results but which are chosen also to avoid having to give technical definitions in this review.

Part I (Corollary 7). Let \(\Gamma\) be a discrete subgroup of \(G=SL(2,R),\) let \(h_ t\) be the horocycle flow on \((\Gamma \setminus G,\mu)\) and let \(h=h_ 1\). Then the number of non-isomorphic ergodic self-joinings of h is infinite if \(\Gamma\) is arithmetic and finite if \(\Gamma\) is not arithmetic. If \(\Gamma\) is maximal and not arithmetic then h has only trivial ergodic self-joinings.

Part II (Corollary 14). Let \(h_ t\) be the horocycle flow on (\(\Gamma\) \(\setminus G,\mu)\) where \(\Gamma\) is maximal and not arithmetic. Then \(h_ 1=h\) is product prime.

There is much more to this paper than these results and we strongly recommend any interested reader to read the description of results, which takes up the first ten pages of the paper, to find out more detailed information on the content and consequences of these results.

Part I (Corollary 7). Let \(\Gamma\) be a discrete subgroup of \(G=SL(2,R),\) let \(h_ t\) be the horocycle flow on \((\Gamma \setminus G,\mu)\) and let \(h=h_ 1\). Then the number of non-isomorphic ergodic self-joinings of h is infinite if \(\Gamma\) is arithmetic and finite if \(\Gamma\) is not arithmetic. If \(\Gamma\) is maximal and not arithmetic then h has only trivial ergodic self-joinings.

Part II (Corollary 14). Let \(h_ t\) be the horocycle flow on (\(\Gamma\) \(\setminus G,\mu)\) where \(\Gamma\) is maximal and not arithmetic. Then \(h_ 1=h\) is product prime.

There is much more to this paper than these results and we strongly recommend any interested reader to read the description of results, which takes up the first ten pages of the paper, to find out more detailed information on the content and consequences of these results.

Reviewer: D.Newton

### MSC:

28D10 | One-parameter continuous families of measure-preserving transformations |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |