Horocycle flow

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

References

[1]	E. Hopf, "Statistik des geodätischen Linien in Manningfaltigkeiten negativer Krümmung" Ber. Verh. Sächs. Akad. Wiss. Leipzig , 91 (1939) pp. 261–304
[2]	O.S. Parasyuk, "Horocycle flows on surfaces of constant negative curvature" Uspekhi Mat. Nauk. , 8 : 3 (1953) pp. 125–126 (In Russian)
[3]	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
[4]	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 MR0393339 Zbl 0256.58009
[5]	B. Marcus, "Unique ergodicity of the horocycle flow: variable negative curvature case" Israel J. Math. , 21 : 2–3 (1975) pp. 133–144 MR0407902 Zbl 0314.58013
[6]	B. Marcus, "Ergodic properties of horocycle flows for surfaces of negative curvature" Ann. of Math. , 105 : 1 (1977) pp. 81–105 MR0458496
[7]	B. Marcus, "The horocycle flow is mixing of all degrees" Invent. Math. , 46 : 3 (1978) pp. 201–209 MR0488168 Zbl 0395.28012
[8a]	L.W. Green, "The generalized geodesic flow" Duke Math. J. , 41 : 1 (1974) pp. 115–126 MR0370659 Zbl 0283.58011 Zbl 0935.53037
[8b]	L.W. Green, "Correction on: The generalized geodesic flow" Duke. Math. J. , 42 (1975) pp. 381
[9]	R. Bowen, "Weak mixing and unique ergodicity on homogeneous spaces" Israel J. Math. , 23 : 3–4 (1976) pp. 267–273 MR0407233 Zbl 0338.43014
[10]	R. Bowen, B. Marcus, "Unique ergodicity for horocycle foliations" Israel J. Math. , 26 : 1 (1977) pp. 43–67 MR0451307 Zbl 0346.58009
[11]	M. Ratner, "Rigidity of horocycle flows" Ann. of Math. , 115 : 3 (1982) pp. 597–614 MR0657240 Zbl 0506.58030