2021, 17: 1-32. doi: 10.3934/jmd.2021001

On mixing and sparse ergodic theorems

1. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received  February 15, 2018 Revised  October 14, 2020 Published  February 2021

We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allow us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis, Shah, and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent of the spectral gap.

Citation: Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001
References:
[1]

J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., 1317, Springer, Berlin, 1988,204–223. doi: 10.1007/BFb0081742.

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45. 

[3]

J. Bourgain, On the Vinogradov integral, Proc. Steklov Inst. Math., 296 (2017), no. 1, 30–40. doi: 10.1134/S0371968517010034.

[4]

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

[5]

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

[6]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 91–137.

[7]

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.

[8]

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.

[9]

L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, preprint, arXiv: 1507.05147. doi: 10.1007/s00039-016-0385-4.

[10]

I. M. Gel'fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Generalized Functions, 6, Academic Press, Inc., Boston, MA, 1990.

[11]

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.

[12]

H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974.

[13]

Ha rish-Chandra, Discrete series for semisimple lie groups. Ⅱ: Explicit determination of the characters, Acta Math., 116 (1966), 1-111.  doi: 10.1007/BF02392813.

[14]

L.-K. Hua, On Waring's problem, The Quarterly Journal of Mathematics, os-9 (1938), 199-202.  doi: 10.1093/qmath/os-9.1.199.

[15]

H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.

[16]

Y. Katznelson, An Introduction to Harmonic Analysis, 3rd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.

[17]

H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 175-181. 

[18]

A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.

[19]

A. W. Knapp, Lie Groups Beyond an Introduction, 2$^nd$ edition, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002.

[20]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.

[21]

G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174.

[22]

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.

[23]

J. M. Rosenblatt and M. Wierdl, Pointwise ergodic theorems via harmonic analysis, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 3–151. doi: 10.1017/CBO9780511574818.002.

[24]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97. 

[25]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618.  doi: 10.1016/j.matpur.2014.07.004.

[26]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732.  doi: 10.1215/S0012-7094-94-07521-2.

[27]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328.  doi: 10.3934/jmd.2013.7.291.

[28]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, Int. Math. Res. Not. IMRN, 24 (2015), 13728-13756.  doi: 10.1093/imrn/rnv115.

[29]

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

[30]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.

[31]

C. Zheng, Sparse equidistribution of unipotent orbits in finite-volume quotients of ${\text PSL}(2, \mathbb{R})$, J. Mod. Dyn., 10 (2016), 1-21.  doi: 10.3934/jmd.2016.10.1.

show all references

References:
[1]

J. Bourgain, An approach to pointwise ergodic theorems, in Geometric Aspects of Functional Analysis (1986/87), Lecture Notes in Math., 1317, Springer, Berlin, 1988,204–223. doi: 10.1007/BFb0081742.

[2]

J. Bourgain, Pointwise ergodic theorems for arithmetic sets, with an appendix by the author, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45. 

[3]

J. Bourgain, On the Vinogradov integral, Proc. Steklov Inst. Math., 296 (2017), no. 1, 30–40. doi: 10.1134/S0371968517010034.

[4]

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

[5]

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

[6]

S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., 16, Amer. Math. Soc., Providence, RI, 1993, 91–137.

[7]

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.

[8]

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.

[9]

L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, preprint, arXiv: 1507.05147. doi: 10.1007/s00039-016-0385-4.

[10]

I. M. Gel'fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Generalized Functions, 6, Academic Press, Inc., Boston, MA, 1990.

[11]

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.

[12]

H. Halberstam and H.-E. Richert, Sieve Methods, London Mathematical Society Monographs, 4, Academic Press, London-New York, 1974.

[13]

Ha rish-Chandra, Discrete series for semisimple lie groups. Ⅱ: Explicit determination of the characters, Acta Math., 116 (1966), 1-111.  doi: 10.1007/BF02392813.

[14]

L.-K. Hua, On Waring's problem, The Quarterly Journal of Mathematics, os-9 (1938), 199-202.  doi: 10.1093/qmath/os-9.1.199.

[15]

H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. doi: 10.1090/coll/053.

[16]

Y. Katznelson, An Introduction to Harmonic Analysis, 3rd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9781139165372.

[17]

H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, J. Amer. Math. Soc., 16 (2003), 175-181. 

[18]

A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.

[19]

A. W. Knapp, Lie Groups Beyond an Introduction, 2$^nd$ edition, Progress in Mathematics, 140, Birkhäuser Boston, Inc., Boston, MA, 2002.

[20]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.

[21]

G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174.

[22]

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.

[23]

J. M. Rosenblatt and M. Wierdl, Pointwise ergodic theorems via harmonic analysis, in Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 3–151. doi: 10.1017/CBO9780511574818.002.

[24]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97. 

[25]

P. Sarnak and A. Ubis, The horocycle flow at prime times, J. Math. Pures Appl. (9), 103 (2015), 575-618.  doi: 10.1016/j.matpur.2014.07.004.

[26]

N. A. Shah, Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J., 75 (1994), 711-732.  doi: 10.1215/S0012-7094-94-07521-2.

[27]

A. Strömbergsson, On the deviation of ergodic averages for horocycle flows, J. Mod. Dyn., 7 (2013), 291-328.  doi: 10.3934/jmd.2013.7.291.

[28]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, Int. Math. Res. Not. IMRN, 24 (2015), 13728-13756.  doi: 10.1093/imrn/rnv115.

[29]

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

[30]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.

[31]

C. Zheng, Sparse equidistribution of unipotent orbits in finite-volume quotients of ${\text PSL}(2, \mathbb{R})$, J. Mod. Dyn., 10 (2016), 1-21.  doi: 10.3934/jmd.2016.10.1.

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