April  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 Published  September 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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

[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.

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

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