Horocyclic flow

In mathematics , the horocyclic flow is an example of an algebraically describable chaotic dynamic system .

definition

Let it be a hyperbolic surface , thus a Riemannian manifold of the form ${\displaystyle F}$

${\displaystyle F=\Gamma \backslash H^{2}}$,

where is the hyperbolic plane and a discrete set of isometrics . ${\displaystyle H^{2}}$ ${\displaystyle \Gamma \subset \operatorname {Isom} (H^{2})}$

The Poincaré model of the hyperbolic plane , various geodesics ending in the same point (in red) and an associated horocycle (in blue).

Consider the hyperbolic plane and its unit tangent bundle . The effect of the group of orientation-maintaining isometries ${\displaystyle H^{2}}$ ${\displaystyle T^{1}H^{2}}$

${\displaystyle \mathrm {Isom} ^{+}(H^{2})\simeq {PSL}(2,\mathbb {R} )}$

on induces a bijection between and . We consider the effect of on as a left effect . Then the horocyclic flow corresponds to the legal effect of on . ${\displaystyle T^{1}H^{2}}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle T^{1}H^{2}}$ ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle T^{1}H^{2}=PSL(2,\mathbb {R} )}$ ${\displaystyle \Psi _{t}}$ ${\displaystyle \left({\begin{array}{cc}1&t\\0&1\end{array}}\right)}$ ${\displaystyle PSL(2,\mathbb {R} )}$

This right- hand action commutes with the left-hand action of , i.e. it induces a well-defined action on the unit tangential bundle ${\displaystyle \Psi _{t}}$ ${\displaystyle \Gamma }$ ${\displaystyle \Psi _{t}}$

${\displaystyle T^{1}F=\Gamma \backslash T^{1}H^{2}}$,

which is called the horocyclic flow .

The orbits of the horocyclic flow are the projections on the area of the constraints of the unit tangential bundle on the horocycles in the hyperbolic plane. ${\displaystyle F}$ ${\displaystyle T^{1}H^{2}}$

properties

Interaction with other rivers

A frequently used property of the horocyclic flow is its interaction with the geodesic flow . It applies ${\displaystyle \Phi _{t}}$

${\displaystyle \Phi _{s}\Psi _{t}\Phi _{-s}=\Psi _{e^{2s}t}}$

for everyone . In particular, the orbits of the horocyclic flow are the stable manifolds of the geodesic flow. ${\displaystyle s,t\in \mathbb {R} }$

Often the so-called negative horocyclic flow is also considered, the effect of which is given by the right effect of on . For this applies ${\displaystyle \Psi _{t}^{-}}$ ${\displaystyle T^{1}H^{2}=PSL(2,\mathbb {R} )}$ ${\displaystyle \left({\begin{array}{cc}1&0\\t&1\end{array}}\right)}$ ${\displaystyle PSL(2,\mathbb {R} )}$

${\displaystyle \Phi _{s}\Psi _{t}^{-}\Phi _{-s}=\Psi _{e^{-2s}t}^{-}}$,

its orbits are the unstable manifolds of the geodesic flow.

Compact areas

If is compact, then the horocyclic flow is minimal , ergodic with respect to the Liouville measure (which in the case of hyperbolic surfaces corresponds to the image of the hair measure under the projection ) and even clearly ergodic, i.e. H. every flow-invariant measure is a scalar multiple of the Liouville measure. In particular, all orbits are equally distributed with regard to the Liouville measure. ${\displaystyle F}$ ${\displaystyle PSL(2,\mathbb {R} )=T^{1}H^{2}\to T^{1}F}$

Non-compact surfaces of finite volume

If it has finite volume (with respect to the hair measure) but is not compact, then one has periodic orbits (corresponding to the closed horocycles around the tips of ), but with the exception of the linear combinations of Dirac measures on these periodic orbits, the scalar ones are Multiples of the Liouville measure are the only river-invariant measures and all non-periodic orbits are evenly distributed with regard to the Liouville measure. ${\displaystyle F}$ ${\displaystyle F}$

literature

• Ghys, Étienne: Dynamique des flots unipotents sur les espaces homogènes. Séminaire Bourbaki, Vol. 1991/92. Astérisque No. 206 (1992), Exp. 747, 3, 93-136.
• Morris, Dave Witte: Ratner's theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2005. ISBN 0-226-53983-0 ; 0-226-53984-9

Individual evidence

1. ↑ Hedlund, Gustav A .: Fuchsian groups and transitive horocycles. Duke Math. J. 2 (1936), no. 3, 530-542.
2. ^ Furstenberg, Harry: The unique ergodicity of the horocycle flow. Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), pp. 95-115. Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973.
3. ^ Dani, SG: Invariant measures of horospherical flows on noncompact homogeneous spaces. Invent. Math. 47 (1978) no. 2, 101-138.
4. ↑ Dani, SG; Smillie, John: Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51 (1984) no. 1, 185-194.