# Hopf-Tsuji-Sullivan theorem

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 |

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Hopf-Tsuji-Sullivan theorem.

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