Invariant measures on the space of horofunctions of a word hyperbolic group
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چکیده
We introduce a natural equivalence relation on the space H0 of horofunctions of a word hyperbolic group that take the value 0 at the identity. We show that there are only finitely many ergodic measures that are invariant under this relation. This can be viewed as a discrete analog of the Bowen-Marcus theorem. Furthermore, if η is such a measure and G acts on a space (X,μ) by p.m.p. transformations then η×μ is virtually ergodic with respect to a natural equivalence relation on H0 ×X. This is comparable to a special case of the Howe-Moore theorem. These results are applied to prove a new ergodic theorem for spherical averages in the case of a word hyperbolic group acting on a finite space.
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تاریخ انتشار 2008