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This theorem establishes the equivalence of several characterizations of  "smallness"  of a [[Riemannian manifold|Riemannian manifold]] of constant negative curvature, or, more generally, of a discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103101.png" /> of isometries of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103102.png" />-dimensional hyperbolic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103103.png" /> (cf. also [[Discrete group of transformations|Discrete group of transformations]]).
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Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103104.png" /> the sphere at infinity (the visibility sphere), of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103105.png" />, and fix an origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103106.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103107.png" /> is called a radial limit point of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103108.png" /> if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h1103109.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031010.png" />-neighbourhood of the geodesic ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031011.png" /> contains infinitely many points from the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031012.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031013.png" /> of all radial limit points is called the radial limit set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031014.png" />. Alternatively, let the shadow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031015.png" /> of the ball <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031016.png" /> of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031017.png" /> centred at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031018.png" /> be the set of end-points of all geodesic rays which are issued from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031019.png" /> and intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031020.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031021.png" /> if and only if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031023.png" /> belongs to an infinite number of shadows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031025.png" />.
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This theorem establishes the equivalence of several characterizations of  "smallness" of a [[Riemannian manifold|Riemannian manifold]] of constant negative curvature, or, more generally, of a discrete group  $  G $
 +
of isometries of the  $  ( d + 1 ) $-
 +
dimensional hyperbolic space  $  \mathbf H ^ {d + 1 } $(
 +
cf. also [[Discrete group of transformations|Discrete group of transformations]]).
 +
 
 +
Denote by  $  \partial  \mathbf H ^ {d + 1 } = S  ^ {d} $
 +
the sphere at infinity (the visibility sphere), of $  \mathbf H ^ {d + 1 } $,  
 +
and fix an origin $  o \in \mathbf H ^ {d + 1 } $.  
 +
A point $  \gamma \in \partial  \mathbf H ^ {d + 1 } $
 +
is called a radial limit point of the group $  G $
 +
if there exists a number $  R > 0 $
 +
such that the $  R $-
 +
neighbourhood of the geodesic ray $  [ o, \gamma ] $
 +
contains infinitely many points from the orbit $  Go = \{ {go } : {g \in G } \} $.  
 +
The set $  \Omega _ {r} \subset  S  ^ {d} $
 +
of all radial limit points is called the radial limit set of $  G $.  
 +
Alternatively, let the shadow $  {\mathcal S} _ {o} ( x,R ) \subset  \partial  \mathbf H ^ {d + 1 } $
 +
of the ball $  B ( x,R ) $
 +
of radius $  R > 0 $
 +
centred at a point $  x \in \mathbf H ^ {d + 1 } $
 +
be the set of end-points of all geodesic rays which are issued from $  o $
 +
and intersect $  B ( x,R ) $.  
 +
Then $  \gamma \in \Omega _ {r} $
 +
if and only if there is an $  R > 0 $
 +
such that $  \gamma $
 +
belongs to an infinite number of shadows $  {\mathcal S} _ {o} ( go,R ) $,  
 +
$  g \in G $.
  
 
The following conditions are equivalent:
 
The following conditions are equivalent:
  
1) The Poincaré series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031026.png" /> diverges, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031027.png" /> is the Riemannian distance on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031028.png" />.
+
1) The Poincaré series $  \sum _ {g \in G }  e ^ {- d  { \mathop{\rm dist} } ( o,go ) } $
 +
diverges, where $  { \mathop{\rm dist} } ( \cdot, \cdot ) $
 +
is the Riemannian distance on $  \mathbf H ^ {d + 1 } $.
  
2) The quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031029.png" /> has no Green function, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031030.png" /> has no non-constant negative subharmonic functions (cf. also [[Subharmonic function|Subharmonic function]]), or, equivalently, the [[Brownian motion|Brownian motion]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031031.png" /> is recurrent.
+
2) The quotient $  M = \mathbf H ^ {d + 1 } /G $
 +
has no Green function, i.e., $  M $
 +
has no non-constant negative subharmonic functions (cf. also [[Subharmonic function|Subharmonic function]]), or, equivalently, the [[Brownian motion|Brownian motion]] on $  M $
 +
is recurrent.
  
3) The complement of the radial limit set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031032.png" /> has [[Lebesgue measure|Lebesgue measure]] zero.
+
3) The complement of the radial limit set $  \partial  \mathbf H ^ {d + 1 } \setminus  \Omega _ {r} $
 +
has [[Lebesgue measure|Lebesgue measure]] zero.
  
4) The [[Geodesic flow|geodesic flow]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031033.png" /> is ergodic with respect to the Liouville-invariant measure (the one determined by the Riemannian volume).
+
4) The [[Geodesic flow|geodesic flow]] on $  M $
 +
is ergodic with respect to the Liouville-invariant measure (the one determined by the Riemannian volume).
  
5) The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031034.png" /> on the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031035.png" /> is ergodic with respect to the [[Lebesgue measure|Lebesgue measure]].
+
5) The action of $  G $
 +
on the product $  \partial  \mathbf H ^ {d + 1 } \times \partial  \mathbf H ^ {d + 1 } $
 +
is ergodic with respect to the [[Lebesgue measure|Lebesgue measure]].
  
Usually the term  "Hopf–Tsuji–Sullivan theorem"  is applied to the equivalence of 1), 3) and 4). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031036.png" /> the implication 3)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031037.png" />4) was first proved by E. Hopf [[#References|[a1]]], [[#References|[a2]]], and the implications 4)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031038.png" />1)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031039.png" />3) by M. Tsuji, see [[#References|[a3]]]. Tsuji's proof is essentially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031040.png" />-dimensional, as it uses complex function theory, whereas Hopf's argument easily carries over to the higher-dimensional case. D. Sullivan [[#References|[a4]]] used an entirely different way for proving the chain of implications 4)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031041.png" />3)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031042.png" />1)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031043.png" />4) for an arbitrary dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031044.png" />.
+
Usually the term  "Hopf–Tsuji–Sullivan theorem"  is applied to the equivalence of 1), 3) and 4). For $  d = 1 $
 +
the implication 3) $  \Rightarrow $
 +
4) was first proved by E. Hopf [[#References|[a1]]], [[#References|[a2]]], and the implications 4) $  \Rightarrow $
 +
1) $  \Rightarrow $
 +
3) by M. Tsuji, see [[#References|[a3]]]. Tsuji's proof is essentially $  2 $-
 +
dimensional, as it uses complex function theory, whereas Hopf's argument easily carries over to the higher-dimensional case. D. Sullivan [[#References|[a4]]] used an entirely different way for proving the chain of implications 4) $  \Rightarrow $
 +
3) $  \Rightarrow $
 +
1) $  \Rightarrow $
 +
4) for an arbitrary dimension $  d $.
  
The equivalence of 1) and 2) follows from the asymptotic equivalence of the Green function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031045.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031046.png" />, whereas the equivalence of 3), 4) and 5) is a much more general fact, see [[Hopf alternative|Hopf alternative]]. Sullivan's idea was to deduce the implication 2)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031047.png" />5) from general properties of recurrent Markov operators. On the other hand, the implication 3)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031048.png" />1) is an easy corollary of the estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031050.png" /> is the image of the Lebesgue measure on the unit tangent sphere at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031051.png" /> under the exponential mapping (a particular case of the Sullivan shadow lemma).
+
The equivalence of 1) and 2) follows from the asymptotic equivalence of the Green function on $  \mathbf H ^ {d + 1 } $
 +
to $  e ^ {- d  { \mathop{\rm dist} } ( x,y ) } $,  
 +
whereas the equivalence of 3), 4) and 5) is a much more general fact, see [[Hopf alternative|Hopf alternative]]. Sullivan's idea was to deduce the implication 2) $  \Rightarrow $
 +
5) from general properties of recurrent Markov operators. On the other hand, the implication 3) $  \Rightarrow $
 +
1) is an easy corollary of the estimate $  \nu _ {o} ( {\mathcal S} ( R,go ) ) \sim e ^ {- d  { \mathop{\rm dist} } ( o,go ) } $,  
 +
where $  \nu _ {o} $
 +
is the image of the Lebesgue measure on the unit tangent sphere at the point $  o $
 +
under the exponential mapping (a particular case of the Sullivan shadow lemma).
  
[[Ergodicity|Ergodicity]] of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031052.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031053.png" /> (i.e., absence of bounded harmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031054.png" />) is weaker than 5). For Riemannian surfaces the implication  "no Green function"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031055.png" />  "no non-constant bounded harmonic functions"  is known as the Myrberg theorem, see [[#References|[a5]]]. In probabilistic terms, this implication can be reformulated as  "ergodicity of the time shift in the bilateral path space"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031056.png" />  "ergodicity of the time shift in the unilateral path space" , or just that recurrence of the [[Brownian motion|Brownian motion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031057.png" /> implies absence of non-constant bounded harmonic functions [[#References|[a6]]]. The latter reformulation allows one to construct examples of discrete groups of isometries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031058.png" /> whose action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031059.png" /> is ergodic and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031060.png" /> is not, in a much simpler way than original Riemann surface examples, [[#References|[a7]]].
+
[[Ergodicity|Ergodicity]] of the action of $  G $
 +
on $  \partial  \mathbf H ^ {d + 1 } $(
 +
i.e., absence of bounded harmonic functions on $  M $)  
 +
is weaker than 5). For Riemannian surfaces the implication  "no Green function"   $ \Rightarrow $"
 +
no non-constant bounded harmonic functions"  is known as the Myrberg theorem, see [[#References|[a5]]]. In probabilistic terms, this implication can be reformulated as  "ergodicity of the time shift in the bilateral path space"   $ \Rightarrow $"
 +
ergodicity of the time shift in the unilateral path space" , or just that recurrence of the [[Brownian motion|Brownian motion]] $  M $
 +
implies absence of non-constant bounded harmonic functions [[#References|[a6]]]. The latter reformulation allows one to construct examples of discrete groups of isometries of $  \mathbf H ^ {d + 1 } $
 +
whose action on $  \partial  \mathbf H ^ {d + 1 } $
 +
is ergodic and on $  \partial  \mathbf H ^ {d + 1 } \times \partial  \mathbf H ^ {d + 1 } $
 +
is not, in a much simpler way than original Riemann surface examples, [[#References|[a7]]].
  
An analogue of the Hopf–Tsuji–Sullivan theorem for the invariant measure of the geodesic flow corresponding to the Patterson–Sullivan measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031061.png" /> was proved in [[#References|[a8]]], see also [[#References|[a9]]]. In this setup, condition 1) is replaced by divergence of the Poincaré series at the critical exponent of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110310/h11031062.png" />.
+
An analogue of the Hopf–Tsuji–Sullivan theorem for the invariant measure of the geodesic flow corresponding to the Patterson–Sullivan measure on $  \partial  \mathbf H ^ {d + 1 } $
 +
was proved in [[#References|[a8]]], see also [[#References|[a9]]]. In this setup, condition 1) is replaced by divergence of the Poincaré series at the critical exponent of the group $  G $.
  
 
In the non-constant curvature case, generalizations of the Hopf–Tsuji–Sullivan theorem were obtained in [[#References|[a6]]] for the harmonic invariant measure of the geodesic flow and in [[#References|[a10]]] for the Patterson–Sullivan measure.
 
In the non-constant curvature case, generalizations of the Hopf–Tsuji–Sullivan theorem were obtained in [[#References|[a6]]] for the harmonic invariant measure of the geodesic flow and in [[#References|[a10]]] for the Patterson–Sullivan measure.

Latest revision as of 22:11, 5 June 2020


This theorem establishes the equivalence of several characterizations of "smallness" of a Riemannian manifold of constant negative curvature, or, more generally, of a discrete group $ G $ of isometries of the $ ( d + 1 ) $- dimensional hyperbolic space $ \mathbf H ^ {d + 1 } $( cf. also Discrete group of transformations).

Denote by $ \partial \mathbf H ^ {d + 1 } = S ^ {d} $ the sphere at infinity (the visibility sphere), of $ \mathbf H ^ {d + 1 } $, and fix an origin $ o \in \mathbf H ^ {d + 1 } $. A point $ \gamma \in \partial \mathbf H ^ {d + 1 } $ is called a radial limit point of the group $ G $ if there exists a number $ R > 0 $ such that the $ R $- neighbourhood of the geodesic ray $ [ o, \gamma ] $ contains infinitely many points from the orbit $ Go = \{ {go } : {g \in G } \} $. The set $ \Omega _ {r} \subset S ^ {d} $ of all radial limit points is called the radial limit set of $ G $. Alternatively, let the shadow $ {\mathcal S} _ {o} ( x,R ) \subset \partial \mathbf H ^ {d + 1 } $ of the ball $ B ( x,R ) $ of radius $ R > 0 $ centred at a point $ x \in \mathbf H ^ {d + 1 } $ be the set of end-points of all geodesic rays which are issued from $ o $ and intersect $ B ( x,R ) $. Then $ \gamma \in \Omega _ {r} $ if and only if there is an $ R > 0 $ such that $ \gamma $ belongs to an infinite number of shadows $ {\mathcal S} _ {o} ( go,R ) $, $ g \in G $.

The following conditions are equivalent:

1) The Poincaré series $ \sum _ {g \in G } e ^ {- d { \mathop{\rm dist} } ( o,go ) } $ diverges, where $ { \mathop{\rm dist} } ( \cdot, \cdot ) $ is the Riemannian distance on $ \mathbf H ^ {d + 1 } $.

2) The quotient $ M = \mathbf H ^ {d + 1 } /G $ has no Green function, i.e., $ M $ has no non-constant negative subharmonic functions (cf. also Subharmonic function), or, equivalently, the Brownian motion on $ M $ is recurrent.

3) The complement of the radial limit set $ \partial \mathbf H ^ {d + 1 } \setminus \Omega _ {r} $ has Lebesgue measure zero.

4) The geodesic flow on $ M $ is ergodic with respect to the Liouville-invariant measure (the one determined by the Riemannian volume).

5) The action of $ G $ on the product $ \partial \mathbf H ^ {d + 1 } \times \partial \mathbf H ^ {d + 1 } $ is ergodic with respect to the Lebesgue measure.

Usually the term "Hopf–Tsuji–Sullivan theorem" is applied to the equivalence of 1), 3) and 4). For $ d = 1 $ the implication 3) $ \Rightarrow $ 4) was first proved by E. Hopf [a1], [a2], and the implications 4) $ \Rightarrow $ 1) $ \Rightarrow $ 3) by M. Tsuji, see [a3]. Tsuji's proof is essentially $ 2 $- dimensional, as it uses complex function theory, whereas Hopf's argument easily carries over to the higher-dimensional case. D. Sullivan [a4] used an entirely different way for proving the chain of implications 4) $ \Rightarrow $ 3) $ \Rightarrow $ 1) $ \Rightarrow $ 4) for an arbitrary dimension $ d $.

The equivalence of 1) and 2) follows from the asymptotic equivalence of the Green function on $ \mathbf H ^ {d + 1 } $ to $ e ^ {- d { \mathop{\rm dist} } ( x,y ) } $, whereas the equivalence of 3), 4) and 5) is a much more general fact, see Hopf alternative. Sullivan's idea was to deduce the implication 2) $ \Rightarrow $ 5) from general properties of recurrent Markov operators. On the other hand, the implication 3) $ \Rightarrow $ 1) is an easy corollary of the estimate $ \nu _ {o} ( {\mathcal S} ( R,go ) ) \sim e ^ {- d { \mathop{\rm dist} } ( o,go ) } $, where $ \nu _ {o} $ is the image of the Lebesgue measure on the unit tangent sphere at the point $ o $ under the exponential mapping (a particular case of the Sullivan shadow lemma).

Ergodicity of the action of $ G $ on $ \partial \mathbf H ^ {d + 1 } $( i.e., absence of bounded harmonic functions on $ M $) is weaker than 5). For Riemannian surfaces the implication "no Green function" $ \Rightarrow $" no non-constant bounded harmonic functions" is known as the Myrberg theorem, see [a5]. In probabilistic terms, this implication can be reformulated as "ergodicity of the time shift in the bilateral path space" $ \Rightarrow $" ergodicity of the time shift in the unilateral path space" , or just that recurrence of the Brownian motion $ M $ implies absence of non-constant bounded harmonic functions [a6]. The latter reformulation allows one to construct examples of discrete groups of isometries of $ \mathbf H ^ {d + 1 } $ whose action on $ \partial \mathbf H ^ {d + 1 } $ is ergodic and on $ \partial \mathbf H ^ {d + 1 } \times \partial \mathbf H ^ {d + 1 } $ is not, in a much simpler way than original Riemann surface examples, [a7].

An analogue of the Hopf–Tsuji–Sullivan theorem for the invariant measure of the geodesic flow corresponding to the Patterson–Sullivan measure on $ \partial \mathbf H ^ {d + 1 } $ was proved in [a8], see also [a9]. In this setup, condition 1) is replaced by divergence of the Poincaré series at the critical exponent of the group $ G $.

In the non-constant curvature case, generalizations of the Hopf–Tsuji–Sullivan theorem were obtained in [a6] for the harmonic invariant measure of the geodesic flow and in [a10] for the Patterson–Sullivan measure.

References

[a1] E. Hopf, "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krummung" Ber. Verh. Sachs. Akad. Wiss. Leipzig , 91 (1939) pp. 261–304
[a2] E. Hopf, "Ergodic theory and the geodesic flow on surfaces of constant negative curvature" Bull. Amer. Math. Soc. , 77 (1971) pp. 863–877
[a3] M. Tsuji, "Potential theory in modern function theory" , Maruzen (1959)
[a4] D. Sullivan, "On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions" Ann. Math. Studies , 97 (1980) pp. 465–496
[a5] L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1960)
[a6] V.A. Kaimanovich, "Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces" J. Reine Angew. Math. , 455 (1994) pp. 57–103
[a7] T. Lyons, D. Sullivan, "Function theory, random paths and covering spaces" J. Diff. Geom. , 19 (1984) pp. 299–323
[a8] D. Sullivan, "The density at infinity of a discrete group of hyperbolic motions" IHES Publ. Math. , 50 (1979) pp. 171–202
[a9] P.J. Nicholls, "Ergodic theory of discrete groups" , Cambridge Univ. Press (1989)
[a10] C.B. Yue, "The ergodic theory of discrete isometry groups on manifolds of variable negative curvature" Trans. Amer. Math. Soc. , 348 (1996) pp. 4965–5005
How to Cite This Entry:
Hopf-Tsuji-Sullivan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf-Tsuji-Sullivan_theorem&oldid=22593
This article was adapted from an original article by V.A. Kaimanovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article