July  2013, 7(2): 291-328. doi: 10.3934/jmd.2013.7.291

On the deviation of ergodic averages for horocycle flows

Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden

Received  April 25, 2013

We give effective bounds on the deviation of ergodic averages for the horocycle flow on the unit tangent bundle of a noncompact hyperbolic surface of finite area. The bounds depend on the small eigenvalues of the Laplacian and on the rate of excursion into cusps for the geodesic corresponding to the given initial point. We also prove Ω-results which show that in a certain sense our bounds are essentially the best possible for any given initial point.

Citation: Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291
References:
[1]

J. S. Athreya and Y. Cheung, A Poincaré section for horocycle flow on the space of lattices, to appear in Int. Math. Res. Not., arXiv: 1206.6597, (2013). doi: 10.1093/imrn/rnt003.  Google Scholar

[2]

J. Bernstein and A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. of Math. (2), 150 (1999), 329–352. doi: 10.2307/121105.  Google Scholar

[3]

A. Borel, "Automorphic Forms on SL2(ℝ), " Cambridge Tracts in Mathematics, 130, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511896064.  Google Scholar

[4]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, arXiv: 1104.4502. Google Scholar

[5]

D. Bump, "Automorphic Forms and Representations, " Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

[6]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[7]

S. G. Dani, On uniformly distributed orbits of certain horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 139-158.   Google Scholar

[8]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194.  doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[9]

M. EinsiedlerG. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.  Google Scholar

[10]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[11]

L. Flaminio and G. Forni, personal communication, 2003. Google Scholar

[12]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[13]

J. L. Hafner, Some Remarks on odd Maass Wave forms (and a correction to "Zeros of L-functions attached to Maass forms" [Math. Z. 190 (1985), no. 1,113–128; MR0793354 (86m:11034)] by Hafner, C. Epstein and P. Sarnak), Math. Z., 196 (1987), 129-132.  doi: 10.1007/BF01179274.  Google Scholar

[14]

D. A. Hejhal, "The Selberg Trace Formula for PSL(2, ℝ). Vol. 2, " Lecture Notes in Math., 1001, Springer-Verlag, Berlin, 1983.  Google Scholar

[15]

H. Iwaniec, "Introduction to the Spectral Theory of Automorphic Forms, " Biblioteca de la Revista Matemática Iberoamericana, Revista Matemática, Iberoamericana, Madrid, 1995.  Google Scholar

[16]

A. Kontorovich and H. Oh, Almost prime Pythagorean Triples In Thin Orbits, J. Reine Angew. Math., 667 (2012), 89-131.   Google Scholar

[17]

S. Lang, "SL(2, ℝ), " Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975.  Google Scholar

[18]

M. Lee and H. Oh, Effective equidistribution of closed horocycles for geometrically finite surfaces, arXiv: 1202.0848. Google Scholar

[19]

J. Lehner, "Discontinuous Groups and Automorphic Functions, " Math. Surveys, No. Ⅷ, American Mathematical Society, Providence, R.I., 1964.  Google Scholar

[20]

G. W. Mackey, "The Theory of Unitary Group Representations, " Chicago Lectures in Mathematics, Univ. of Chicago Press, Chicage, Ill.-London, 1976.  Google Scholar

[21]

G. A. Margulis, Problems and conjectures in rigidity theory, in "Mathematics: Frontiers and Perspectives, " AMS, Providence, RI, (2000), 161–174.  Google Scholar

[22]

F. Maucourant and B. Weiss, Lattice actions on the plane revisited, Geom. Dedicata, 157 (2012), 1-21.  doi: 10.1007/s10711-011-9596-x.  Google Scholar

[23]

M. V. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93.  doi: 10.1307/mmj/1029004675.  Google Scholar

[24]

T. Miyake, "Modular Forms, " Springer-Verlag, Berlin, 1989.  Google Scholar

[25]

E. Nelson, Analytic vectors, Ann. of Math. (2), 70 (1959), 572–615. doi: 10.2307/1970331.  Google Scholar

[26]

S. J. Patterson, Diophantine approximation in Fuchsian groups, Phil. Trans. Soc. London Ser. A, 282 (1976), 527-563.  doi: 10.1098/rsta.1976.0063.  Google Scholar

[27]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[28]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.  doi: 10.1215/S0012-7094-91-06311-8.  Google Scholar

[29]

M. Ratner, Raghunathan's conjectures for SL(2, ℝ), Israel J. Math., 80 (1992), 1–31. doi: 10.1007/BF02808152.  Google Scholar

[30]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739.  doi: 10.1002/cpa.3160340602.  Google Scholar

[31]

P. Sarnak and A. Ubis, The horocycle flow at prime times, arXiv: 1110.0777. Google Scholar

[32]

H. Shimizu, On discontinuous groups acting on the product of the upper half planes, Ann. Math. (2), 77 (1963), 33–71. doi: 10.2307/1970201.  Google Scholar

[33]

A. Strömbergsson, On the uniform equidistribution of long closed horocycles, Duke Math. J., 123 (2004), 507-547.  doi: 10.1215/S0012-7094-04-12334-6.  Google Scholar

[34]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

[35]

S. L. Velani, Diophantine approximation and Hausdorff dimension in Fuchsian groups, Math. Proc. Cambridge Philos. Soc., 113 (1993), 343-354.  doi: 10.1017/S0305004100076015.  Google Scholar

[36]

S. L. Velani, Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension, Math. Proc. Cambridge Philos. Soc., 120 (1996), 647-662.  doi: 10.1017/S0305004100001626.  Google Scholar

[37]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989–1094. doi: 10.4007/annals.2010.172.989.  Google Scholar

[38]

N. R. Wallach, "Real Reductive Groups. Ⅰ, Ⅱ, " Academic Press, Inc., Boston, MA, 1988, 1992.  Google Scholar

[39]

G. N. Watson, "A Treatise on the Theory of Bessel Functions, " Cambridge Univ. Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

show all references

References:
[1]

J. S. Athreya and Y. Cheung, A Poincaré section for horocycle flow on the space of lattices, to appear in Int. Math. Res. Not., arXiv: 1206.6597, (2013). doi: 10.1093/imrn/rnt003.  Google Scholar

[2]

J. Bernstein and A. Reznikov, Analytic continuation of representations and estimates of automorphic forms, Ann. of Math. (2), 150 (1999), 329–352. doi: 10.2307/121105.  Google Scholar

[3]

A. Borel, "Automorphic Forms on SL2(ℝ), " Cambridge Tracts in Mathematics, 130, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511896064.  Google Scholar

[4]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, arXiv: 1104.4502. Google Scholar

[5]

D. Bump, "Automorphic Forms and Representations, " Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997. doi: 10.1017/CBO9780511609572.  Google Scholar

[6]

M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803.  doi: 10.1215/S0012-7094-90-06129-0.  Google Scholar

[7]

S. G. Dani, On uniformly distributed orbits of certain horocycle flows, Ergodic Theory Dynam. Systems, 2 (1982), 139-158.   Google Scholar

[8]

S. G. Dani and J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J., 51 (1984), 185-194.  doi: 10.1215/S0012-7094-84-05110-X.  Google Scholar

[9]

M. EinsiedlerG. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212.  doi: 10.1007/s00222-009-0177-7.  Google Scholar

[10]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J., 119 (2003), 465-526.  doi: 10.1215/S0012-7094-03-11932-8.  Google Scholar

[11]

L. Flaminio and G. Forni, personal communication, 2003. Google Scholar

[12]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2), 175 (2012), 465–540. doi: 10.4007/annals.2012.175.2.2.  Google Scholar

[13]

J. L. Hafner, Some Remarks on odd Maass Wave forms (and a correction to "Zeros of L-functions attached to Maass forms" [Math. Z. 190 (1985), no. 1,113–128; MR0793354 (86m:11034)] by Hafner, C. Epstein and P. Sarnak), Math. Z., 196 (1987), 129-132.  doi: 10.1007/BF01179274.  Google Scholar

[14]

D. A. Hejhal, "The Selberg Trace Formula for PSL(2, ℝ). Vol. 2, " Lecture Notes in Math., 1001, Springer-Verlag, Berlin, 1983.  Google Scholar

[15]

H. Iwaniec, "Introduction to the Spectral Theory of Automorphic Forms, " Biblioteca de la Revista Matemática Iberoamericana, Revista Matemática, Iberoamericana, Madrid, 1995.  Google Scholar

[16]

A. Kontorovich and H. Oh, Almost prime Pythagorean Triples In Thin Orbits, J. Reine Angew. Math., 667 (2012), 89-131.   Google Scholar

[17]

S. Lang, "SL(2, ℝ), " Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975.  Google Scholar

[18]

M. Lee and H. Oh, Effective equidistribution of closed horocycles for geometrically finite surfaces, arXiv: 1202.0848. Google Scholar

[19]

J. Lehner, "Discontinuous Groups and Automorphic Functions, " Math. Surveys, No. Ⅷ, American Mathematical Society, Providence, R.I., 1964.  Google Scholar

[20]

G. W. Mackey, "The Theory of Unitary Group Representations, " Chicago Lectures in Mathematics, Univ. of Chicago Press, Chicage, Ill.-London, 1976.  Google Scholar

[21]

G. A. Margulis, Problems and conjectures in rigidity theory, in "Mathematics: Frontiers and Perspectives, " AMS, Providence, RI, (2000), 161–174.  Google Scholar

[22]

F. Maucourant and B. Weiss, Lattice actions on the plane revisited, Geom. Dedicata, 157 (2012), 1-21.  doi: 10.1007/s10711-011-9596-x.  Google Scholar

[23]

M. V. Melián and D. Pestana, Geodesic excursions into cusps in finite-volume hyperbolic manifolds, Michigan Math. J., 40 (1993), 77-93.  doi: 10.1307/mmj/1029004675.  Google Scholar

[24]

T. Miyake, "Modular Forms, " Springer-Verlag, Berlin, 1989.  Google Scholar

[25]

E. Nelson, Analytic vectors, Ann. of Math. (2), 70 (1959), 572–615. doi: 10.2307/1970331.  Google Scholar

[26]

S. J. Patterson, Diophantine approximation in Fuchsian groups, Phil. Trans. Soc. London Ser. A, 282 (1976), 527-563.  doi: 10.1098/rsta.1976.0063.  Google Scholar

[27]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

[28]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280.  doi: 10.1215/S0012-7094-91-06311-8.  Google Scholar

[29]

M. Ratner, Raghunathan's conjectures for SL(2, ℝ), Israel J. Math., 80 (1992), 1–31. doi: 10.1007/BF02808152.  Google Scholar

[30]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739.  doi: 10.1002/cpa.3160340602.  Google Scholar

[31]

P. Sarnak and A. Ubis, The horocycle flow at prime times, arXiv: 1110.0777. Google Scholar

[32]

H. Shimizu, On discontinuous groups acting on the product of the upper half planes, Ann. Math. (2), 77 (1963), 33–71. doi: 10.2307/1970201.  Google Scholar

[33]

A. Strömbergsson, On the uniform equidistribution of long closed horocycles, Duke Math. J., 123 (2004), 507-547.  doi: 10.1215/S0012-7094-04-12334-6.  Google Scholar

[34]

D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., 149 (1982), 215-237.  doi: 10.1007/BF02392354.  Google Scholar

[35]

S. L. Velani, Diophantine approximation and Hausdorff dimension in Fuchsian groups, Math. Proc. Cambridge Philos. Soc., 113 (1993), 343-354.  doi: 10.1017/S0305004100076015.  Google Scholar

[36]

S. L. Velani, Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension, Math. Proc. Cambridge Philos. Soc., 120 (1996), 647-662.  doi: 10.1017/S0305004100001626.  Google Scholar

[37]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2), 172 (2010), 989–1094. doi: 10.4007/annals.2010.172.989.  Google Scholar

[38]

N. R. Wallach, "Real Reductive Groups. Ⅰ, Ⅱ, " Academic Press, Inc., Boston, MA, 1988, 1992.  Google Scholar

[39]

G. N. Watson, "A Treatise on the Theory of Bessel Functions, " Cambridge Univ. Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

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