# 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 of isometries of the -dimensional hyperbolic space (cf. also Discrete group of transformations).

Denote by the sphere at infinity (the visibility sphere), of , and fix an origin . A point is called a radial limit point of the group if there exists a number such that the -neighbourhood of the geodesic ray contains infinitely many points from the orbit . The set of all radial limit points is called the radial limit set of . Alternatively, let the shadow of the ball of radius centred at a point be the set of end-points of all geodesic rays which are issued from and intersect . Then if and only if there is an such that belongs to an infinite number of shadows , .

The following conditions are equivalent:

1) The Poincaré series diverges, where is the Riemannian distance on .

2) The quotient has no Green function, i.e., has no non-constant negative subharmonic functions (cf. also Subharmonic function), or, equivalently, the Brownian motion on is recurrent.

3) The complement of the radial limit set has Lebesgue measure zero.

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

5) The action of on the product 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 the implication 3)4) was first proved by E. Hopf [a1], [a2], and the implications 4)1)3) by M. Tsuji, see [a3]. Tsuji's proof is essentially -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)3)1)4) for an arbitrary dimension .

The equivalence of 1) and 2) follows from the asymptotic equivalence of the Green function on to , 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)5) from general properties of recurrent Markov operators. On the other hand, the implication 3)1) is an easy corollary of the estimate , where is the image of the Lebesgue measure on the unit tangent sphere at the point under the exponential mapping (a particular case of the Sullivan shadow lemma).

Ergodicity of the action of on (i.e., absence of bounded harmonic functions on ) is weaker than 5). For Riemannian surfaces the implication "no Green function" "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" "ergodicity of the time shift in the unilateral path space" , or just that recurrence of the Brownian motion implies absence of non-constant bounded harmonic functions [a6]. The latter reformulation allows one to construct examples of discrete groups of isometries of whose action on is ergodic and on 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 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 .

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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