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Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. (English) Zbl 0566.58022
Let $$\Gamma$$ 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.
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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