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Checking atomicity of conformal ending measures for Kleinian groups. (English) Zbl 1218.30121

The basic idea of a conformal ending measure is as follows. Let \((z_n)\) be an ending sequence, that is, a sequence of points in the Poincaré ball model \((\mathbb D, d)\) of the hyperbolic space which converges radially to some point \(\zeta\in\mathbb S =\partial\mathbb D\) within some Dirichlet fundamental domain of a Kleinian group \(G\). If \(\zeta\) is a limit point of \(G\), then the closure of the orbit \(G(z_n)\) converges to the limit set \(L(G)\) for \(n\) tending to infinity. Here, convergence is meant with respect to the Hausdorff metric on closed subsets of the closure \(\overline{\mathbb D} =\mathbb D\cup\mathbb S\) in the Euclidean topology. If \(s\in\mathbb R^+\) is chosen such that the Poincaré series \(\sum_{g\in G}j(g,z)^s\) of \(G\) at \(z\in\mathbb D\) with exponent \(s\) converges, then there exist purely atomic \(s\)-conformal probability measures \(\mu_n\) having their atoms in \(G(z_n)\). Here, \(j(g,z)\) denotes the conformal derivative of \(g\) at \(z\in\overline{\mathbb D}\), that is, the uniquely determined positive number such that \(g'(z)/j(g,z)\) is orthogonal, where \(g'(z)\) is the Jacobian of \(g\) at \(z\). For \(\zeta\in\mathbb S\) we have
\[ j(g,\zeta) = \frac{1 -|g^{-1}(0)|^2}{|\zeta - g^{-1}(0)|^2}. \]
Consequently, the weak limits of these measures \(\mu_n\) give rise to \(s\)-conformal measures supported on \(L(G)\). These measures are called \(s\)-conformal ending measures.
The aim of this paper is to address questions of continuity and atomicity for this type of ending measures. Throughout the phrase ‘essential support of a measure’ is used to denote a measurable set of full measure (not necessarily closed and not uniquely determined). The Dirac delta measure at \(z\) is denoted by \(\mathbf{1}_z\). The following statement summarizes the results of the paper.
Let \((z_n)\) be a given ending sequence converging to \(\zeta\in\mathbb S\), and assume that \(j(h,\zeta) = 1\) for every element \(h\) of the stabilizer \(\text{Stab}_G(\zeta)\) of \(\zeta\). Moreover, assume that the reduced horospherical Poincaré series
\[ \sum_{g\in G/\text{Stab}_G(\zeta)}j(g,\zeta)^s \]
of \(G\) at \(\zeta\) with exponent \(s\) converges.
Then the essential support of the associated \(s\)-conformal ending measure \(\mu\) is equal to \(G(\zeta)\). In particular, \(\mu\) is purely atomic and coincides with the measure \(\mu_\zeta\) given by
\[ \mu_\zeta=\frac{\sum_{g\in G/\text{Stab}_G(\zeta)}j(g,\zeta)^s\mathbf{1}_{g(\zeta)}}{\sum_{g\in G/\text{Stab}_G(\zeta)}j(g,\zeta)^s}. \]
If \(\zeta\) is an ordinary point of \(G\) or a bounded parabolic fixed point of \(G\), then the reduced horospherical Poincaré series of \(G\) at \(\zeta\) with exponent \(s\in\mathbb R^+\) converges whenever the Poincaré series \(\sum_{g\in G}j(g,z)^s\), \(z\in\mathbb D\), converges.
This result provides typical examples of purely atomic conformal ending measures. Another class of examples is obtained by using a result of J. W. Anderson, K. Falk and P. Tukia [Q. J. Math. 58, No. 1, 1–15 (2007; Zbl 1167.30021)], stating that a conformal ending measure associated to a certain given end is essentially supported on the associated end limit set. We then have that if this end limit set is countable, then every ending measure essentially supported on it is necessarily purely atomic. Besides, this raises the question whether it is possible to have a countable end limit set associated to an end which does not originate from a parabolic fixed point. In fact, a result of C. Bishop [Ann. Acad. Sci. Fenn., Math. 21, No. 2, 383–388 (1996; Zbl 0849.30034)], stating that a Kleinian group acting on three-dimensional hyperbolic space is geometrically finite if and only if the set of non-radial limit points is countable, strongly suggests that the case of parabolic cusps is the only case in which the end limit set of a given end is countable.
The authors first show that if a conformal ending measure for a Kleinian group \(G\) has an atom which is contained in the big horospherical limit set \(L_H(G)\), then this atom has to be a parabolic fixed point. Note that the above mentioned result of the authors gives a sufficient condition, in terms of the convergence of the reduced horospherical Poincaré series at \(\zeta\), under which a conformal ending measure is purely atomic. However, unless \(\zeta\) is an ordinary point or a bounded parabolic fixed point, it is usually not an easy task to verify this convergence condition. Hence it is desirable to find conditions which allow to decide more easily whether a conformal ending measure has atoms or not. By employing a result of P. Tukia [Ann. Acad. Sci. Fenn., Math. 22, No. 2, 387–394 (1997; Zbl 0890.30029)], which states that there always exists a measurable fundamental set for the action of a Kleinian group \(G\) on the complement of \(L_H(G)\), it is shown that if \(G\) admits an \(s\)-conformal measure essentially supported on the dissipative part of the action of \(G\) on \(\mathbb S\), then we always have that there exists a purely atomic \(s\)-conformal ending measure for \(G\). This observation then gives rise to the following result, where \(\delta(G)\) denotes the abscissa of convergence of the Poincaré series of \(G\).
For \(s\geq \delta(G)\), every \(s\)-conformal ending measure for \(G\) supported on \(L(G)\) is essentially supported on \(L_H(G)\) if and only if every \(s\)-conformal ending measure for \(G\) supported on \(L(G)\) can have atoms only at parabolic fixed points of \(G\).
Eventually, the authors give various non-trivial examples of purely atomic and non-atomic conformal ending measures. These include examples of infinitely generated Schottky groups whose conformal ending measures have atoms at Jørgensen points, and examples of non-atomic conformal ending measures on limit sets of normal subgroups of finitely generated Schottky groups. Moreover, the construction of a Kleinian group \(\Gamma\) is given, which admits a purely atomic \(\delta(\Gamma)\)-conformal ending measure at some Jørgensen point, which is not a parabolic fixed point of \(\Gamma\).

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
28A80 Fractals
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References:

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