2017, 11: 189-217. doi: 10.3934/jmd.2017009

Effective equidistribution of circles in the limit sets of Kleinian groups

Mathematics Department, Yale University, New Haven, CT 06520, USA

Received  November 25, 2015 Revised  October 22, 2016 Published  February 2017

Consider a general circle packing $\mathscr{P}$ in the complex plane $\mathbb{C}$ invariant under a Kleinian group $\Gamma$. When $\Gamma$ is convex cocompact or its critical exponent is greater than 1, we obtain an effective equidistribution for small circles in $\mathscr{P}$ intersecting any bounded connected regular set in $\mathbb{C}$; this provides an effective version of an earlier work of Oh-Shah [12]. In view of the recent result of McMullen-Mohammadi-Oh [6], our effective circle counting theorem applies to the circles contained in the limit set of a convex cocompact but non-cocompact Kleinian group whose limit set contains at least one circle. Moreover, consider the circle packing $\mathscr{P}(\mathscr{T})$ of the ideal triangle attained by filling in largest inner circles. We give an effective estimate to the number of disks whose hyperbolic areas are greater than $t$, as $t\to0$, effectivizing the work of Oh [10].

Citation: Wenyu Pan. Effective equidistribution of circles in the limit sets of Kleinian groups. Journal of Modern Dynamics, 2017, 11: 189-217. doi: 10.3934/jmd.2017009
References:
[1]

C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718. Google Scholar

[2]

R. D. Canary and E. Taylor, Kleinian groups with small limit sets, Duke Math J., 73 (1994), 371-381. doi: 10.1215/S0012-7094-94-07316-X. Google Scholar

[3]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. With an appendix by Oh and Nimish Shah, J. Amer. Math. Soc., 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7. Google Scholar

[4]

M. Lee and H. Oh, Effective count for Apollonian circle packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8. Google Scholar

[5]

C. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721. doi: 10.1353/ajm.1998.0031. Google Scholar

[6]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, H. Invent. Math. , (2016). Available from: http://gauss.math.yale.edu/~ho2/. doi: 10.1007/s00222-016-0711-3. Google Scholar

[7]

A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc.(JEMS), 17 (2015), 837-897. doi: 10.4171/JEMS/520. Google Scholar

[8]

A. Mohammadi and H. Oh, Ergodicity of unipotent flows and Kleinian groups, J. Amer. Math. Soc., 28 (2015), 531-577. doi: 10.1090/S0894-0347-2014-00811-0. Google Scholar

[9] D. MumfordC. Series and D. Wright, Indra's Pearls. The Vision of Felix Klein, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9781107050051.024. Google Scholar
[10]

H. Oh, Harmonic analysis, ergodic theory and counting for thin groups, in Thin Groups and Superstrong Approximation (eds. E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ., 61, Cambridge Univ. Press., Cambridge, 2014. Google Scholar

[11]

H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyper bolic groups, J. Amer. Math. Soc., 26 (2013), 511-562. doi: 10.1090/S0894-0347-2012-00749-8. Google Scholar

[12]

H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math., 187 (2012), 1-35. doi: 10.1007/s00222-011-0326-7. Google Scholar

[13]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, To appear in Ergodic Theory and Dynamical Systems, arXiv: 1305.1332. doi: 10.1017/etds.2015.77. Google Scholar

[14]

P. Vesselin and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. Google Scholar

[15]

B. Stratmann and M. Urbański, Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Philos. Soc., 140 (2006), 297-304. doi: 10.1017/S0305004105009114. Google Scholar

[16]

B. Stratmann and S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements. new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220. doi: 10.1112/plms/s3-71.1.197. Google Scholar

[17]

D. Sullivan, Entropy, Hausdorff measures old and new. and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379. Google Scholar

[18]

L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. Google Scholar

[19]

I. Vinogradov, Effective bisector estimate with applications to Apollonian circle packings, Int. Math. Res. Not. IMRN 2014,3217–3262. Google Scholar

show all references

References:
[1]

C. J. Bishop and P. W. Jones, Hausdorff dimension and Kleinian groups, Acta Math., 179 (1997), 1-39. doi: 10.1007/BF02392718. Google Scholar

[2]

R. D. Canary and E. Taylor, Kleinian groups with small limit sets, Duke Math J., 73 (1994), 371-381. doi: 10.1215/S0012-7094-94-07316-X. Google Scholar

[3]

A. Kontorovich and H. Oh, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds. With an appendix by Oh and Nimish Shah, J. Amer. Math. Soc., 24 (2011), 603-648. doi: 10.1090/S0894-0347-2011-00691-7. Google Scholar

[4]

M. Lee and H. Oh, Effective count for Apollonian circle packings and closed horospheres, Geom. Funct. Anal., 23 (2013), 580-621. doi: 10.1007/s00039-013-0217-8. Google Scholar

[5]

C. McMullen, Hausdorff dimension and conformal dynamics. Ⅲ. Computation of dimension, Amer. J. Math., 120 (1998), 691-721. doi: 10.1353/ajm.1998.0031. Google Scholar

[6]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in hyperbolic 3-manifolds, H. Invent. Math. , (2016). Available from: http://gauss.math.yale.edu/~ho2/. doi: 10.1007/s00222-016-0711-3. Google Scholar

[7]

A. Mohammadi and H. Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc.(JEMS), 17 (2015), 837-897. doi: 10.4171/JEMS/520. Google Scholar

[8]

A. Mohammadi and H. Oh, Ergodicity of unipotent flows and Kleinian groups, J. Amer. Math. Soc., 28 (2015), 531-577. doi: 10.1090/S0894-0347-2014-00811-0. Google Scholar

[9] D. MumfordC. Series and D. Wright, Indra's Pearls. The Vision of Felix Klein, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9781107050051.024. Google Scholar
[10]

H. Oh, Harmonic analysis, ergodic theory and counting for thin groups, in Thin Groups and Superstrong Approximation (eds. E. Breulliard and H. Oh), Math. Sci. Res. Inst. Publ., 61, Cambridge Univ. Press., Cambridge, 2014. Google Scholar

[11]

H. Oh and N. Shah, Equidistribution and counting for orbits of geometrically finite hyper bolic groups, J. Amer. Math. Soc., 26 (2013), 511-562. doi: 10.1090/S0894-0347-2012-00749-8. Google Scholar

[12]

H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math., 187 (2012), 1-35. doi: 10.1007/s00222-011-0326-7. Google Scholar

[13]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, To appear in Ergodic Theory and Dynamical Systems, arXiv: 1305.1332. doi: 10.1017/etds.2015.77. Google Scholar

[14]

P. Vesselin and L. Stoyanov, Distribution of periods of closed trajectories in exponentially shrinking intervals, Comm. Math. Phys., 310 (2012), 675-704. doi: 10.1007/s00220-012-1419-x. Google Scholar

[15]

B. Stratmann and M. Urbański, Diophantine extremality of the Patterson measure, Math. Proc. Cambridge Philos. Soc., 140 (2006), 297-304. doi: 10.1017/S0305004105009114. Google Scholar

[16]

B. Stratmann and S. L. Velani, The Patterson measure for geometrically finite groups with parabolic elements. new and old, Proc. London Math. Soc. (3), 71 (1995), 197-220. doi: 10.1112/plms/s3-71.1.197. Google Scholar

[17]

D. Sullivan, Entropy, Hausdorff measures old and new. and limit sets of geometrically finite Kleinian groups, Acta Math., 153 (1984), 259-277. doi: 10.1007/BF02392379. Google Scholar

[18]

L. Stoyanov, Spectra of Ruelle transfer operators for axiom A flows, Nonlinearity, 24 (2011), 1089-1120. doi: 10.1088/0951-7715/24/4/005. Google Scholar

[19]

I. Vinogradov, Effective bisector estimate with applications to Apollonian circle packings, Int. Math. Res. Not. IMRN 2014,3217–3262. Google Scholar

Figure 1.  Circle packing intersecting bounded region (background pictures are reproduced from Indra's Pearls: The Vision of Felix Klein, by D. Mumford, C. Series and D. Wright, copyright Cambridge University Press 2002)
Figure 2.  Circle packing in ideal hyperbolic triangle
Figure 3.  Apollonian circle packing $\mathscr{P}$ and generators of $\mathcal{A}$
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