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Keywords
(6)
Asymptotic Distribution
borel measure
Circle Packing
Critical Exponent
Fixed Point
kleinian group
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The asymptotic distribution of circles in the orbits of Kleinian groups
The asymptotic distribution of circles in the orbits of Kleinian groups,10.1007/s0022201103267,Inventiones Mathematicae,Hee Oh,Nimish Shah
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The asymptotic distribution of circles in the orbits of Kleinian groups
(
Citations: 2
)
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Hee Oh
,
Nimish Shah
Let be a locally finite
circle packing
in the plane ℂ invariant under a nonelementary
Kleinian group
Γ and with finitely many Γorbits. When Γ is geometrically finite, we construct an explicit
Borel measure
on ℂ which describes the
asymptotic distribution
of small circles in , assuming that either the
critical exponent
of Γ is strictly bigger than 1 or does not contain an infinite bouquet of tangent circles glued at a parabolic
fixed point
of Γ. Our construction also works for invariant under a geometrically infinite group Γ, provided Γ admits a finite BowenMargulisSullivan measure and the Γskinning size of is finite. Some concrete circle packings to which our result applies include Apollonian circle packings, Sierpinski curves, Schottky dances, etc.
Journal:
Inventiones Mathematicae  INVENT MATH
, vol. 187, no. 1, pp. 13, pp. 135, 2012
DOI:
10.1007/s0022201103267
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References
(16)
Periodic Points and Measures for Axiom a Diffeomorphisms
(
Citations: 86
)
Rufus Bowen
Journal:
Transactions of The American Mathematical Society  TRANS AMER MATH SOC
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The residual set dimension of the Apollonian packing
(
Citations: 22
)
David W. Boyd
Journal:
Mathematika
, vol. 20, no. 02, 1973
Horocycle flow on geometrically finite surfaces
(
Citations: 25
)
Marc Burger
Journal:
Duke Mathematical Journal  DUKE MATH J
, vol. 61, no. 1990, pp. 779803, 1990
Topological immersions of peanian continua in a spherical surface
(
Citations: 16
)
S. Claytor
Strong wavefront lemma and counting lattice points in sectors
(
Citations: 4
)
Alexander Gorodnik
,
Hee Oh
,
Nimish Shah
Journal:
Israel Journal of Mathematics  ISR J MATH
, vol. 176, no. 1, pp. 419444, 2010
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Citations
(2)
Equidistribution and Counting for orbits of geometrically finite hyperbolic groups
Hee Oh
,
Nimish Shah
Published in 2012.
Counting visible circles on the sphere and Kleinian groups
(
Citations: 1
)
Hee Oh
,
Nimish Shah
Published in 2010.