Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups.

*(English)*Zbl 0566.58022Let \(\Gamma\) be a geometrically finite Kleinian group acting in hyperbolic space \({\mathbb{H}}^ 3\) (or more generally \({\mathbb{H}}^ n)\) with limit set \(\Lambda\). This paper continues the study begun earlier [D. Sullivan, Publ. Math., Inst. Hautes Etud. Sci. 50, 171-202 (1979; Zbl 0439.30034)] of geometric measures on \(\Lambda\). These are finite measures \(\mu\) which translate by the rule \(\gamma *\mu =| \gamma '|^{\delta}\mu\), \(\gamma\in \Gamma\), where \(\delta =Haus. Dim.(\Lambda)\). (\(\delta\) is also the critical exponent \(\overline{\lim}_{R\to \infty}(1/R)\log n(R)\) of \(\Gamma\), where n(R) is the number of orbit points in a ball of radius R and fixed center.) Such \(\mu\) are constructed following Patterson as a limit of weighted atomic measures on orbits of \(\Gamma\) in \({\mathbb{H}}^ 3\). The author proves that \(\mu\) is uniquely determined by the translation rule and \(\mu (\Lambda)=1.\)

Let \(\nu_ c\) denote the Hausdorff (covering) measure of \(\lambda\) with gauge function \(r^{\delta}\). Dually one introduces the packing measure \(\nu_ p(U)=\lim_{\epsilon \to 0} \sup \sum r_ i^{\delta}\), where \(U\subset \Lambda\) is open and the sup is over packings of U by disjoint balls \(B_ i\in U\) of radii \(\leq \epsilon\). Let \(\mu\) (x,r) be the size of a ball of center x, radius r. The inequalities \[ c\leq \mu (x,r)/r^{\delta}\leq_{i.o.}C\quad and\quad c\leq_{i.o.}\mu (x,r)/r^{\delta}\leq C \] (where i.o. means for a sequence \(r_ i\to 0)\) imply respectively that \(\mu =\nu_ p\) and \(\mu =\nu_ c.\)

Suitable estimates show that if \(\Gamma\) has no cusps then \(\mu =\nu_ p=\nu_ c\). If \(\Gamma\) has cusps the situation is more complicated: one may have \(\mu =\nu_ c\neq \nu_ p\) (rank 1 cusps, \(1<\delta <2)\) or \(\mu =\nu_ p\neq \nu_ c\) (rank 1 cusps, \(\delta <1)\). The only unresolved case is when \(1<\delta <2\) and \(\Gamma\) has cusps of ranks 1 and 2. Then \(0=\nu_ c\neq \nu_ p=\infty\) and \(\mu\) is unidentified.

One constructs an invariant measure \(m_{\mu}\) for the geodesic flow on \(T_ 1({\mathbb{H}}^ 3/\Gamma)\) as the product of \(\mu \times \mu /| x- y|^{2\delta}\) on pairs of endpoints of geodesics on the sphere at infinity with arc length along geodesics. This is important in proving uniqueness of \(\mu\). Also, \(m_{\mu}\) has finite total mass and is ergodic with measure theoretic entropy \(\delta\). In the case of no cusps, \(\delta\) is also the topological entropy.

Let \(\nu_ c\) denote the Hausdorff (covering) measure of \(\lambda\) with gauge function \(r^{\delta}\). Dually one introduces the packing measure \(\nu_ p(U)=\lim_{\epsilon \to 0} \sup \sum r_ i^{\delta}\), where \(U\subset \Lambda\) is open and the sup is over packings of U by disjoint balls \(B_ i\in U\) of radii \(\leq \epsilon\). Let \(\mu\) (x,r) be the size of a ball of center x, radius r. The inequalities \[ c\leq \mu (x,r)/r^{\delta}\leq_{i.o.}C\quad and\quad c\leq_{i.o.}\mu (x,r)/r^{\delta}\leq C \] (where i.o. means for a sequence \(r_ i\to 0)\) imply respectively that \(\mu =\nu_ p\) and \(\mu =\nu_ c.\)

Suitable estimates show that if \(\Gamma\) has no cusps then \(\mu =\nu_ p=\nu_ c\). If \(\Gamma\) has cusps the situation is more complicated: one may have \(\mu =\nu_ c\neq \nu_ p\) (rank 1 cusps, \(1<\delta <2)\) or \(\mu =\nu_ p\neq \nu_ c\) (rank 1 cusps, \(\delta <1)\). The only unresolved case is when \(1<\delta <2\) and \(\Gamma\) has cusps of ranks 1 and 2. Then \(0=\nu_ c\neq \nu_ p=\infty\) and \(\mu\) is unidentified.

One constructs an invariant measure \(m_{\mu}\) for the geodesic flow on \(T_ 1({\mathbb{H}}^ 3/\Gamma)\) as the product of \(\mu \times \mu /| x- y|^{2\delta}\) on pairs of endpoints of geodesics on the sphere at infinity with arc length along geodesics. This is important in proving uniqueness of \(\mu\). Also, \(m_{\mu}\) has finite total mass and is ergodic with measure theoretic entropy \(\delta\). In the case of no cusps, \(\delta\) is also the topological entropy.

Reviewer: C.Series

##### MSC:

37A99 | Ergodic theory |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

54F50 | Topological spaces of dimension \(\leq 1\); curves, dendrites |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

53D25 | Geodesic flows in symplectic geometry and contact geometry |

11F06 | Structure of modular groups and generalizations; arithmetic groups |

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##### References:

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