Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are ...characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.
Let
O
be a closed
n
-dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of
O
is bounded above by
c
1
log
vol
(
O
)
+
c
2
h
(
O
)
,
where
h
(
O
)
is the Cheeger ...constant of
O
,
vol
(
O
)
is its volume, and constants
c
1
,
c
2
depend only on
n
.
Let O be a hyperbolic 3-orbifold with underlying space the 3-sphere. If O contains an essential 2-suborbifold with underlying space the 2-sphere with four cone points, we show how to compute the guts ...of O split along the 2-suborbifold. When the guts are non-empty, we obtain volume bounds in terms of the topology of the guts. When the guts are empty, we give a complete description of the topological structure of O.
THE SMALLEST HAKEN HYPERBOLIC POLYHEDRA ATKINSON, CHRISTOPHER K.; RAFALSKI, SHAWN
Proceedings of the American Mathematical Society,
04/2013, Letnik:
141, Številka:
4
Journal Article
Recenzirano
Odprti dostop
We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3—orbifold contains an essential 2—suborbifold, up to a canonical decomposition along essential ...hyperbolic triangle 2—suborbifolds.
We detect the cusped complete hyperbolic orbifolds of minimal volume in dimensions less than ten. Our method is geometric and uses results from crystallography and the theory of sphere packings. In ...fact, such an
n-orbifold gives rise to a horoball packing whose orthogonal projection to the cusp boundary yields the densest euclidean lattice packing in dimension
n
−
1
.
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