doi: 10.3934/dcds.2022057
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Limit theorems for higher rank actions on Heisenberg nilmanifolds

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

*Corresponding author: Minsung Kim

Received  March 2021 Revised  December 2021 Early access April 2022

Fund Project: The first author is supported by NSF grant DMS 1600687 and by the Centre of Excellence "Dynamics, mathematical analysis and artificial intelligence" at Nicolaus Copernicus University in Toruń

The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct Bufetov functionals on rectangles on $ (2g+1) $-dimensional Heisenberg manifolds. We prove that deviation of the ergodic integral of higher rank actions is described by the asymptotic of Bufetov functionals for a sufficiently smooth function. As a corollary, the distribution of normalized ergodic integrals which have variance 1, converges along certain subsequences to a non-degenerate compactly supported measure on the real line.

Citation: Minsung Kim. Limit theorems for higher rank actions on Heisenberg nilmanifolds. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022057
References:
[1]

A. Adam and V. Baladi, Horocycle averages on closed manifolds and transfer operators,, preprint, 2018, arXiv: 1809.04062.

[2]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[3]

A. AvilaG. Forni and C. Ulcigrai, Mixing for the time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. 

[4]

V. Baladi, There are no deviations for the ergodic averages of Giulietti-Liverani horocycle flows on the two-torus, Ergodic Theory Dynam. Systems, 42 (2022), 500-513.  doi: 10.1017/etds.2021.17.

[5]

A. Brudnyi, On local behavior of analytic functions, J. Funct. Anal., 169 (1999), 481-493.  doi: 10.1006/jfan.1999.3481.

[6]

A. Bufetov, Hölder cocycles and ergodic integrals for translation flows on flat surfaces, Electron. Res. Announc. Math. Sci., 17 (2010), 34-42.  doi: 10.3934/era.2010.17.34.

[7]

____, Limit theorems for suspension flows over Vershik automorphisms, Russian Mathematical Surveys, 68 (2013), 789.

[8]

A. I. Bufetov, Finitely-additive measures on the asymptotic foliations of a markov compactum, Mosc. Math. J., 14 (2014), 205-224.  doi: 10.17323/1609-4514-2014-14-2-205-224.

[9]

A. I. Bufetov, Limit theorems for translation flows, Ann. of Math., 179 (2014), 431-499.  doi: 10.4007/annals.2014.179.2.2.

[10]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47 (2014), 851-903.  doi: 10.24033/asens.2229.

[11]

A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789.  doi: 10.1007/s00220-012-1624-7.

[12]

O. Butterley and L. Simonelli, Parabolic flows renormalized by partially hyperbolic maps, Boll. Unione Mat. Ital., 13 (2020), 341-360.  doi: 10.1007/s40574-020-00235-8.

[13]

F. Cellarosi, J. Griffin and T. Osman, Improved tail estimates for the distribution of quadratic weyl sums,, preprint, 2022, arXiv: 2203.06274.

[14]

F. Cellarosi and J. Marklof, Quadratic weyl sums, automorphic functions and invariance principles, Proc. Lond. Math. Soc., 113 (2016), 775-828.  doi: 10.1112/plms/pdw038.

[15]

S. Cosentino and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, J. Mod. Dyn., 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.

[16]

D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713.  doi: 10.1007/s10955-016-1689-3.

[17]

F. FaureS. Gouëzel and E. Lanneau, Ruelle spectrum of linear pseudo-anosov maps, J. Éc. polytech. Math., 6 (2019), 811-877.  doi: 10.5802/jep.107.

[18]

H. Fiedler, W. Jurkat, and O. Körner, Asymptotic expansions of finite theta series., Acta Arithmetica 32.2 (1977): 129-146.

[19]

F. Faure and M. Tsujii, Prequantum Transfer Operator for Symplectic Anosov Diffeomorphism, Astérisque, 2015.

[20]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433.  doi: 10.1017/S014338570500060X.

[21]

____, On effective equidistribution for higher step nilflows, preprint, 2014, arXiv: 1407.3640.

[22]

L. FlaminioG. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geom. Funct. Anal., 26 (2016), 1359-1448.  doi: 10.1007/s00039-016-0385-4.

[23]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.  doi: 10.2307/3062150.

[24]

____, Effective Equidistribution of Nilflows and Bounds on Weyl Sums., Dynamics and Analytic Number Theory, 437 (2016): 136.

[25]

____, On the equidistribution of unstable curves for pseudo-anosov diffeomorphisms of compact surfaces, Ergodic Theory Dynam. Systems, 42 (2020), 855–880. doi: 10.1017/etds.2021.119.

[26]

____, Ruelle resonances from cohomological equations, preprint, (2020), arXiv: 2007.03116.

[27]

G. Forni and A. Kanigowski, Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows, J. Éc. Polytech. Math., 7 (2020), 63-91.  doi: 10.5802/jep.111.

[28]

____, Time-changes of heisenberg nilflows, Asterisque, 416 (2020), 253–299

[29]

P. Giulietti and C. Liverani, Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc., 21 (2019), 2793-2858.  doi: 10.4171/JEMS/892.

[30]

A. Gorodnik, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. 

[31]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphisms, J. Anal. Math., 123 (2014), 355-396.  doi: 10.1007/s11854-014-0024-7.

[32]

F. Götze and M. Gordin, Limiting distributions of theta series on siegel half-spaces, St. Petersburg Math. J., 15 (2004), 81-102.  doi: 10.1090/S1061-0022-03-00803-3.

[33]

J. Griffin and J. Marklof, Limit theorems for skew translations, J. Mod. Dyn., 8 (2014), 177-189.  doi: 10.3934/jmd.2014.8.177.

[34]

A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, No. 30. American Mathematical Soc., 2003. doi: 10.1090/ulect/030.

[35]

A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory and Dynamical Systems, 15 (1995), 569-592.  doi: 10.1017/S0143385700008531.

[36]

M. Kim, Effective equidistribution for generalized higher step nilflows, Ergodic Theory and Dynamical Systems, (2021), 1–60. doi: 10.1017/etds.2021.110.

[37]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[38]

J. Marklof, Limit theorems for theta sums, Duke Math. J., 97 (1999), 127-153.  doi: 10.1215/S0012-7094-99-09706-5.

[39]

S. MarmiP. Moussa and J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100 (2010), 639-669.  doi: 10.1112/plms/pdp037.

[40]

S. MarmiC. Ulcigrai and J.-C. Yoccoz, On Roth type conditions, duality and central Birkhoff sums for iem, Astérisque, 416 (2020), 65-132. 

[41]

D. Mumford and C. Musili, Tata Lectures on Theta. i (modern Birkhäuser Classics), Birkhäuser Boston Incorporated, 2007.

[42]

D. Mumford, M. Nori and P. Norman, Tata Lectures on Theta iii, Birkhäuser Boston, Inc., Boston, MA, 2007.

[43]

D. Ravotti, Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows, Ergodic Theory Dynam. Systems, 39 (2019), 3407-3436.  doi: 10.1017/etds.2018.19.

[44]

____, Asymptotics and limit theorems for horocycle ergodic integrals a la Ratner, preprint, (2021), arXiv: 2107.02090.

[45]

N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279.  doi: 10.1215/00127094-2009-026.

[46]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 281-304.  doi: 10.1215/00127094-2009-027.

[47]

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.

[48]

R. Tolimieri, Heisenberg manifolds and theta functions, Trans. Amer. Math. Soc., 239 (1978), 293-319.  doi: 10.1090/S0002-9947-1978-0487050-7.

[49]

T.D Wooley., Perturbations of Weyl sums, International Mathematics Research Notices 2016.9 (2015): 2632-2646.

show all references

References:
[1]

A. Adam and V. Baladi, Horocycle averages on closed manifolds and transfer operators,, preprint, 2018, arXiv: 1809.04062.

[2]

A. Avila, G. Forni, D. Ravotti and C. Ulcigrai, Mixing for smooth time-changes of general nilflows, Adv. Math., 385 (2021), 107759, 65 pp. doi: 10.1016/j.aim.2021.107759.

[3]

A. AvilaG. Forni and C. Ulcigrai, Mixing for the time-changes of heisenberg nilflows, J. Differential Geom., 89 (2011), 369-410. 

[4]

V. Baladi, There are no deviations for the ergodic averages of Giulietti-Liverani horocycle flows on the two-torus, Ergodic Theory Dynam. Systems, 42 (2022), 500-513.  doi: 10.1017/etds.2021.17.

[5]

A. Brudnyi, On local behavior of analytic functions, J. Funct. Anal., 169 (1999), 481-493.  doi: 10.1006/jfan.1999.3481.

[6]

A. Bufetov, Hölder cocycles and ergodic integrals for translation flows on flat surfaces, Electron. Res. Announc. Math. Sci., 17 (2010), 34-42.  doi: 10.3934/era.2010.17.34.

[7]

____, Limit theorems for suspension flows over Vershik automorphisms, Russian Mathematical Surveys, 68 (2013), 789.

[8]

A. I. Bufetov, Finitely-additive measures on the asymptotic foliations of a markov compactum, Mosc. Math. J., 14 (2014), 205-224.  doi: 10.17323/1609-4514-2014-14-2-205-224.

[9]

A. I. Bufetov, Limit theorems for translation flows, Ann. of Math., 179 (2014), 431-499.  doi: 10.4007/annals.2014.179.2.2.

[10]

A. Bufetov and G. Forni, Limit theorems for horocycle flows, Ann. Sci. Éc. Norm. Supér., 47 (2014), 851-903.  doi: 10.24033/asens.2229.

[11]

A. Bufetov and B. Solomyak, Limit theorems for self-similar tilings, Comm. Math. Phys., 319 (2013), 761-789.  doi: 10.1007/s00220-012-1624-7.

[12]

O. Butterley and L. Simonelli, Parabolic flows renormalized by partially hyperbolic maps, Boll. Unione Mat. Ital., 13 (2020), 341-360.  doi: 10.1007/s40574-020-00235-8.

[13]

F. Cellarosi, J. Griffin and T. Osman, Improved tail estimates for the distribution of quadratic weyl sums,, preprint, 2022, arXiv: 2203.06274.

[14]

F. Cellarosi and J. Marklof, Quadratic weyl sums, automorphic functions and invariance principles, Proc. Lond. Math. Soc., 113 (2016), 775-828.  doi: 10.1112/plms/pdw038.

[15]

S. Cosentino and L. Flaminio, Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds, J. Mod. Dyn., 9 (2015), 305-353.  doi: 10.3934/jmd.2015.9.305.

[16]

D. Dolgopyat and O. Sarig, Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., 166 (2017), 680-713.  doi: 10.1007/s10955-016-1689-3.

[17]

F. FaureS. Gouëzel and E. Lanneau, Ruelle spectrum of linear pseudo-anosov maps, J. Éc. polytech. Math., 6 (2019), 811-877.  doi: 10.5802/jep.107.

[18]

H. Fiedler, W. Jurkat, and O. Körner, Asymptotic expansions of finite theta series., Acta Arithmetica 32.2 (1977): 129-146.

[19]

F. Faure and M. Tsujii, Prequantum Transfer Operator for Symplectic Anosov Diffeomorphism, Astérisque, 2015.

[20]

L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory Dynam. Systems, 26 (2006), 409-433.  doi: 10.1017/S014338570500060X.

[21]

____, On effective equidistribution for higher step nilflows, preprint, 2014, arXiv: 1407.3640.

[22]

L. FlaminioG. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geom. Funct. Anal., 26 (2016), 1359-1448.  doi: 10.1007/s00039-016-0385-4.

[23]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math., 155 (2002), 1-103.  doi: 10.2307/3062150.

[24]

____, Effective Equidistribution of Nilflows and Bounds on Weyl Sums., Dynamics and Analytic Number Theory, 437 (2016): 136.

[25]

____, On the equidistribution of unstable curves for pseudo-anosov diffeomorphisms of compact surfaces, Ergodic Theory Dynam. Systems, 42 (2020), 855–880. doi: 10.1017/etds.2021.119.

[26]

____, Ruelle resonances from cohomological equations, preprint, (2020), arXiv: 2007.03116.

[27]

G. Forni and A. Kanigowski, Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows, J. Éc. Polytech. Math., 7 (2020), 63-91.  doi: 10.5802/jep.111.

[28]

____, Time-changes of heisenberg nilflows, Asterisque, 416 (2020), 253–299

[29]

P. Giulietti and C. Liverani, Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc., 21 (2019), 2793-2858.  doi: 10.4171/JEMS/892.

[30]

A. Gorodnik, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. 

[31]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphisms, J. Anal. Math., 123 (2014), 355-396.  doi: 10.1007/s11854-014-0024-7.

[32]

F. Götze and M. Gordin, Limiting distributions of theta series on siegel half-spaces, St. Petersburg Math. J., 15 (2004), 81-102.  doi: 10.1090/S1061-0022-03-00803-3.

[33]

J. Griffin and J. Marklof, Limit theorems for skew translations, J. Mod. Dyn., 8 (2014), 177-189.  doi: 10.3934/jmd.2014.8.177.

[34]

A. Katok, Combinatorial Constructions in Ergodic Theory and Dynamics, No. 30. American Mathematical Soc., 2003. doi: 10.1090/ulect/030.

[35]

A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory and Dynamical Systems, 15 (1995), 569-592.  doi: 10.1017/S0143385700008531.

[36]

M. Kim, Effective equidistribution for generalized higher step nilflows, Ergodic Theory and Dynamical Systems, (2021), 1–60. doi: 10.1017/etds.2021.110.

[37]

D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494.  doi: 10.1007/s002220050350.

[38]

J. Marklof, Limit theorems for theta sums, Duke Math. J., 97 (1999), 127-153.  doi: 10.1215/S0012-7094-99-09706-5.

[39]

S. MarmiP. Moussa and J.-C. Yoccoz, Affine interval exchange maps with a wandering interval, Proc. Lond. Math. Soc., 100 (2010), 639-669.  doi: 10.1112/plms/pdp037.

[40]

S. MarmiC. Ulcigrai and J.-C. Yoccoz, On Roth type conditions, duality and central Birkhoff sums for iem, Astérisque, 416 (2020), 65-132. 

[41]

D. Mumford and C. Musili, Tata Lectures on Theta. i (modern Birkhäuser Classics), Birkhäuser Boston Incorporated, 2007.

[42]

D. Mumford, M. Nori and P. Norman, Tata Lectures on Theta iii, Birkhäuser Boston, Inc., Boston, MA, 2007.

[43]

D. Ravotti, Mixing for suspension flows over skew-translations and time-changes of quasi-abelian filiform nilflows, Ergodic Theory Dynam. Systems, 39 (2019), 3407-3436.  doi: 10.1017/etds.2018.19.

[44]

____, Asymptotics and limit theorems for horocycle ergodic integrals a la Ratner, preprint, (2021), arXiv: 2107.02090.

[45]

N. Shah, Limiting distributions of curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 251-279.  doi: 10.1215/00127094-2009-026.

[46]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 281-304.  doi: 10.1215/00127094-2009-027.

[47]

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.

[48]

R. Tolimieri, Heisenberg manifolds and theta functions, Trans. Amer. Math. Soc., 239 (1978), 293-319.  doi: 10.1090/S0002-9947-1978-0487050-7.

[49]

T.D Wooley., Perturbations of Weyl sums, International Mathematics Research Notices 2016.9 (2015): 2632-2646.

Figure 1.  Illustration of the rectangles $ U({\textbf{T}_1}) $ and $ U({\textbf{T}_2}) $ on $ i,j $-th coordinate
Figure 2.  Illustration of the standard $ d $-rectangles $ \Gamma $, $ \Gamma_Q $, $ d+1 $ dimensional current $ D(\Gamma, \Gamma_ \mathsf{Q}) $ and supports of $ r_{-t}(\Gamma) $ and $ r_{-t}(\Gamma_ \mathsf{Q}) $
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