Patterson-Sullivan measure

Limes set of a discrete group of isometries of 3-dimensional hyperbolic space.

In mathematics , Patterson-Sullivan measures are an aid to investigate discrete groups of isometries of symmetrical spaces by means of their "dynamics in infinity". These are measures on the limit set of the group with certain equivariance and absolute continuity properties.

They were first introduced for the hyperbolic level by Samuel Patterson and for higher-dimensional hyperbolic spaces by Dennis Sullivan and generalized by Paul Albuquerque for symmetrical spaces of the non-compact type .

definition

Let be a locally symmetric space with a Riemannian metric . Then there is a family of probability measures on the edge at infinity with the following properties ${\displaystyle M=\Gamma \backslash X}$ ${\displaystyle g}$ ${\displaystyle \left\{\mu _{x}\right\}_{x\in X}}$ ${\displaystyle \partial X}$

• All are atomic measures . ${\displaystyle \mu _{x}}$
• The family of measures is - equivariant : ${\displaystyle \left\{\mu _{x}\right\}}$ ${\displaystyle \Gamma }$

${\displaystyle \gamma _{*}\mu _{x}=\mu _{\gamma x}\ \forall \gamma \in \Gamma }$.

• For everyone is absolutely constant regarding . The Radon-Nikodým derivative is given by ${\displaystyle x,y\in X}$ ${\displaystyle \mu _{y}}$ ${\displaystyle \mu _{x}}$

${\displaystyle {\frac {d\mu _{x}}{d\mu _{y}}}(\xi )=e^{h(g)B(x,y,\xi )}}$,

where is the volume entropy and the Busemann function . ${\displaystyle h(g)}$ ${\displaystyle B(x,y,\xi )}$

construction

For and consider the Poincaré series ${\displaystyle x\in X}$ ${\displaystyle s\in \mathbb {R} }$

${\displaystyle \psi (s)=\sum _{\gamma \in \Gamma }e^{-sd(x,\gamma x)}}$.

There is a “critical exponent” so that the series converges for and diverges for. ${\displaystyle \delta (\Gamma )}$ ${\displaystyle s>\delta (\Gamma )}$ ${\displaystyle s<\delta (\Gamma )}$

Patterson-Sullivan measures are obtained for monotonically decreasing sequences as a weak - * - accumulation point of the sequence of measures ${\displaystyle \mu _{x}}$ ${\displaystyle s\to \delta (\Gamma )}$

${\displaystyle \mu _{s,x}={\frac {1}{\psi (s)}}\sum _{\gamma \in \Gamma }e^{-sd(x,\gamma x)}D_{\gamma x}}$,

where the Dirac measure in designated and the case is added thereto still a slowly increasing function. ${\displaystyle D_{\gamma x}}$ ${\displaystyle \gamma x}$ ${\displaystyle \psi (\delta (\Gamma ))<\infty }$

Applications

Potential theory

For a Patterson-Sullivan measure and a constrained function on , one obtains a harmonic function on through ${\displaystyle \left\{\mu _{x}\right\}_{x\in X}}$ ${\displaystyle f}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle X}$

${\displaystyle F(x)=\int _{\partial _{\infty }X}fd\mu _{x}}$.

It applies . ${\displaystyle \Vert F\Vert _{\infty }=\Vert f\Vert _{\infty }}$

Rigidity Theorems

For the proof of different rigidity theorems it is useful to use an edge map

${\displaystyle f\colon \partial _{\infty }X\to \partial _{\infty }}$

a canonical map

${\displaystyle F\colon X\to Y}$

to be able to construct with. ${\displaystyle \partial _{\infty }F=f}$

To do this, consider the embedding given by the Patterson-Sullivan measures

${\displaystyle X\to {\mathcal {M}}(\partial _{\infty }X)}$,

the push forward

${\displaystyle f_{*}\colon {\mathcal {M}}(\partial _{\infty }X)\to {\mathcal {M}}(\partial _{\infty }Y)}$

and the barycenter

${\displaystyle bar\colon {\mathcal {M}}(\partial _{\infty }y)\to Y}$,

and defined as the sequential execution of these figures. ${\displaystyle F}$

Small groups

Defined after identification ${\displaystyle \partial _{\infty }H^{n}=S^{n-1}}$

${\displaystyle \nu :={\frac {\mu \times \mu }{\vert x-y\vert ^{2\delta }}}}$

with the chordal distance an -invariant dimension on the complement of the diagonals in . Since this space can be identified with the unit tangential bundle, a measure gives up which is invariant under the geodetic flow . The geodesic flow is either ergodic or dissipative for this measure. If so , then the geodetic flow is ergodic if and only if the Poincaré series diverges in the critical value . ${\displaystyle \vert x-y\vert }$ ${\displaystyle \Gamma }$ ${\displaystyle S^{n-1}\times S^{n-1}}$ ${\displaystyle T^{1}H^{n}}$ ${\displaystyle \nu }$ ${\displaystyle T^{1}M}$ ${\displaystyle \delta (\Gamma )>{\tfrac {n-1}{2}}}$ ${\displaystyle s=\delta (\Gamma )}$

For convex-co-compact groups , the Hausdorff dimension is the Limes set. The Poincaré series diverges in the critical value . ${\displaystyle \delta (\Gamma )}$ ${\displaystyle s=\delta (\Gamma )}$

properties

From the first condition and the transitivity of on it follows that all are probability measures . So you get an embedding of in the space of the probability measures on the edge at infinity. ${\displaystyle Isom(X)}$ ${\displaystyle X}$ ${\displaystyle \mu _{x}}$ ${\displaystyle X}$

If is a Zariski-dense subgroup of , then there exists an orbit such that the carrier of each Patterson-Sullivan measure is the average of that orbit with the Limes set of . If also , then there is only one Patterson-Sullivan measure. ${\displaystyle \Gamma }$ ${\displaystyle Isom(X)}$ ${\displaystyle G}$ ${\displaystyle \partial _{\infty }X}$ ${\displaystyle \Gamma }$ ${\displaystyle \psi (\delta (\Gamma ))=\infty }$

If a grating in , then the carrier is of the barycenter of a Weyl chamber at infinity. In general, however, for Zariski-dense groups, every orbit of a regular point can occur as a carrier of . ${\displaystyle \Gamma }$ ${\displaystyle Isom(M)}$ ${\displaystyle \mu _{x}}$ ${\displaystyle G}$ ${\displaystyle \mu _{x}}$

The carrier of is therefore contained in a subset ${\displaystyle \mu _{x}}$

${\displaystyle \partial _{F}X\subset \partial _{\infty }X}$,

which is equivariant isomorphic to the Furstenberg boundary . is the only -invariant probability measure on . ${\displaystyle \mu _{x}}$ ${\displaystyle Stab(x)}$ ${\displaystyle \partial _{F}X}$

literature

• SJ Patterson, The limit set of a Fuchsian group. Acta Math. 136: 241-273 (1976).
• D. Sullivan, The density at infinity of a discrete group of hyperbolic motions. IHES Publ. Math. 50 (1979), 171-202.
• G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. (GAFA) 7: 755-782 (1997).
• P. Albuquerque, Patterson-Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. (GAFA), Vol. 9 (1999), 1-28.

Web links

• J.-F. Quint : An overview of Patterson-Sullivan theory

Individual evidence

1. ↑ G.Besson, G.Courtois, S.Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement negative, Geom. Funct. Anal. 5 (1995), no. 5, 731-799.
2. ↑ C. Connell, B. Farb, The Degree Theorem in Higher Rank, J. Diff. Geom., Vol. 65 (2003), 19-59.