an:06709160
Zbl 1379.37090
Jeon, Woojin; Ohshika, Ken'ichi
Measurable rigidity for Kleinian groups
EN
Ergodic Theory Dyn. Syst. 36, No. 8, 2498-2511 (2016).
00361232
2016
j
37F30 30F40 28C10 57M60
Kleinian group; Patterson-Sullivan measure; rigidity theorem; Cannon-Thurston map
Deformation and rigidity are very important topics in the theory of Kleinian groups. \textit{D. Sullivan} [ Ann. Math. Stud. 97, 465--496 (1981; Zbl 0567.58015)] proved that there are no quasi-conformal deformations supported on limit sets, which is analogous to Mostow rigidity. Moreover, he proved a measurable rigidity theorem and \textit{P. Tukia} [Invent. Math. 97, No. 2, 405--431 (1989; Zbl 0674.30038)] extended this rigidity in a more general setting.
Let \(G\) and \(H\) be Kleinian groups, and \(\mu_G\) and \(\mu_H\) conformal measures, where \(\mu_G\) has no atom. Suppose that at least one of the dimensions \(\mu_G\) and \(\mu_H\) is positive. Suppose that there is an essential injective, measurable and essentially directly measurable (i.e., the image of any measurable set outside some fixed \(\mu_G\)-null set is measurable) map \(f:\Lambda_G\to \Lambda_H\) which conjugates \(G\) to \(H\) almost everywhere. Then either there is a set \(A\) in \(\Lambda_G\) of full measure with \(\mu_H(f(A))=0\) or \(f\) coincides with the restriction of a conformal automorphism to \(\Lambda_G\) almost everywhere and the dimensions of \(\mu_G\) and \(\mu_H\) coincide.
However the existence of such map \(f\) is not evident. The authors prove that an equivariant map from the limit set of \(G\) to that of \(H\) is of divergence type.
Tao Chen (Long Island City)
Zbl 0567.58015; Zbl 0674.30038