View source for Horocycle flow

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A flow in the space of bihedra of an $n$-dimensional Riemannian manifold $M^n$ (usually closed) for which the concept of a horocycle is defined; the horocycle flow describes the movement of the bihedra along the horocycles which they define.

The basic cases in which the concept of a horocycle is defined are those in which the curvature of the Riemannian metric is negative, and either $n=2$ or the curve is constant. With a bihedron, i.e. an orthonormal $2$-frame $(x,e_1,e_2)$ ($x\in M^n$; $e_1,e_2$ are mutually orthogonal unit tangent vectors at the point $x$) is associated the horocycle $h(x,e_1,e_2)$ through $x$ in the direction of $e_2$. It is situated on the horosphere $H(x,e_1)$ through $x$, the $(n-1)$-dimensional manifold orthogonal to the family of geodesic lines, asymptotic (in the positive direction) to the geodesic line which passes through $x$ in the direction of $e_1$. The direction on $h$ defined by $e_2$ is taken to be positive (in case $n=2$ this is the only role of $e_2$; $H$ and $h$ can have self-intersections; the simplest way to avoid the ambiguity which can arise from this is to carry out analogous constructions not in $M^n$, but in its universal covering manifold — when the curvature is constant, this is the ordinary $n$-dimensional Lobachevskii space — and to project the horocycle obtained there into $M^n$). Under the action of a horocycle flow, the bihedron $(x,e_1,e_2)$ during time $t$ passes to

$$(x(t),e_1(t),e_2(t)),$$

where $x(t)$, when $t$ increases, moves at unit velocity along $h(x,e_1,e_2)$ in the positive direction, the unit vector $e_1(t)$ is orthogonal to $H(x,e_1)$ at the point $x(t)$ (the choice of one of the two possible directions for $e_1(t)$ is made by continuity) and $e_2(t)=dx(t)/dt$.

Horocycle flows were studied because they played an important role in the research of geodesic flows (cf. [[Geodesic flow|Geodesic flow]]) on manifolds of negative curvature [[#References|[1]]]. Now this role is played by certain foliations (cf. [[Foliation|Foliation]]) arising from the theory of $Y$-systems (cf. [[Y-system|$Y$-system]]), and horocycle flows became a research subject in its own right. The properties of a horocycle flow have been well established (see [[#References|[2]]]–[[#References|[7]]], [[#References|[11]]]). For various generalizations, see –[[#References|[10]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Hopf, "Statistik des geodätischen Linien in Manningfaltigkeiten negativer Krümmung" ''Ber. Verh. Sächs. Akad. Wiss. Leipzig'' , '''91''' (1939) pp. 261–304</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> O.S. Parasyuk, "Horocycle flows on surfaces of constant negative curvature" ''Uspekhi Mat. Nauk.'' , '''8''' : 3 (1953) pp. 125–126 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.M. Gurevich, "The entropy of horocycle flows" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 124–126 ''Dokl. Akad. Nauk. SSSR'' , '''136''' : 4 (1961) pp. 768–770</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> H. Furstenberg, "The unique ergodicity of the horocycle flow" A. Beck (ed.) , ''Recent advances in topological dynamics'' , ''Lect. notes in math.'' , '''318''' , Springer (1973) pp. 95–115 {{MR|0393339}} {{ZBL|0256.58009}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B. Marcus, "Unique ergodicity of the horocycle flow: variable negative curvature case" ''Israel J. Math.'' , '''21''' : 2–3 (1975) pp. 133–144 {{MR|0407902}} {{ZBL|0314.58013}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B. Marcus, "Ergodic properties of horocycle flows for surfaces of negative curvature" ''Ann. of Math.'' , '''105''' : 1 (1977) pp. 81–105 {{MR|0458496}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B. Marcus, "The horocycle flow is mixing of all degrees" ''Invent. Math.'' , '''46''' : 3 (1978) pp. 201–209 {{MR|0488168}} {{ZBL|0395.28012}} </TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top"> L.W. Green, "The generalized geodesic flow" ''Duke Math. J.'' , '''41''' : 1 (1974) pp. 115–126 {{MR|0370659}} {{ZBL|0283.58011}} {{ZBL|0935.53037}} </TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top"> L.W. Green, "Correction on: The generalized geodesic flow" ''Duke. Math. J.'' , '''42''' (1975) pp. 381</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> R. Bowen, "Weak mixing and unique ergodicity on homogeneous spaces" ''Israel J. Math.'' , '''23''' : 3–4 (1976) pp. 267–273 {{MR|0407233}} {{ZBL|0338.43014}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R. Bowen, B. Marcus, "Unique ergodicity for horocycle foliations" ''Israel J. Math.'' , '''26''' : 1 (1977) pp. 43–67 {{MR|0451307}} {{ZBL|0346.58009}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> M. Ratner, "Rigidity of horocycle flows" ''Ann. of Math.'' , '''115''' : 3 (1982) pp. 597–614 {{MR|0657240}} {{ZBL|0506.58030}} </TD></TR></table>