# 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
How to Cite This Entry:
Hopf-Tsuji-Sullivan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf-Tsuji-Sullivan_theorem&oldid=47266
This article was adapted from an original article by V.A. Kaimanovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article