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キーワード:
Mathematics, Geometric Topology,Dynamical Systems,
要旨:
In this paper, we study the topological behavior of elementary planes in the
Apollonian orbifold $M_A$, whose limit set is the classical Apollonian gasket.
The existence of these elementary planes leads to the following failure of
equidistribution: there exists a sequence of closed geodesic planes in $M_A$
limiting only on a finite union of closed geodesic planes. This contrasts with
other acylindrical hyperbolic 3-manifolds analyzed in [MMO1, arXiv:1802.03853,
arXiv:1802.04423].
On the other hand, we show that certain rigidity still holds: the area of an
elementary plane in $M_A$ is uniformly bounded above, and the union of all
elementary planes is closed. This is achieved by obtaining a complete list of
elementary planes in $M_A$, indexed by their intersection with the convex core
boundary. The key idea is to recover information on a closed geodesic plane in
$M_A$ from its boundary data; requiring the plane to be elementary in turn puts
restrictions on these data.
