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Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k

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Abstract

Let X = G/Γ be a homogeneous space with ambient group G containing the group H = (SO(n, 1))k and x ∈ X be such that Hx is dense in X. Given an analytic curve φ: I = [a, b] → H, we will show that if φ satisfies certain geometric condition, then for a typical diagonal subgroup A = {a(t): t φ ℝ}⊂ H the translates {a(t)φ(I)x: t > 0 of the curve φ(I)x will tend to be equidistributed in X as t → +∞. The proof is based on Ratner’s theorem and linearization technique.

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References

  1. Aka, M., Breuillard, E., Rosenzweig, L., et al.: On metric Diophantine approximation in matrices and Lie groups. Comptes Rendus Mathematique, 353(3), 185–189 (2015)

    Article  MathSciNet  Google Scholar 

  2. Aka, M., Breuillard, E., Rosenzweig, L., et al.: Diophantine approximation on matrices and Lie groups. Geom. Funct. Anal., 28(1), 1–57 (2018)

    Article  MathSciNet  Google Scholar 

  3. Baker, R. C.: Dirichlet’s theorem on Diophantine approximation. Math. Proc. Cambridge Philos. Soc., 83(1), 37–59 (1978)

    Article  MathSciNet  Google Scholar 

  4. Bugeaud, Y.: Approximation by algebraic integers and Hausdorff dimension. Journal of the London Mathematical Society, 65(3), 547–559 (2002)

    Article  MathSciNet  Google Scholar 

  5. Dani, S. G.: On orbits of unipotent flows on homogeneous spaces. Ergodic Theory and Dynamical Systems, 4(1), 25–34 (1984)

    Article  MathSciNet  Google Scholar 

  6. Dani, S. G., Margulis, G. A.: Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci., 101(1), 1–17 (1991)

    Article  MathSciNet  Google Scholar 

  7. Davenport, H., Schmidt, W.: Dirichlet’s theorem on diophantine approximation. II. Acta Arithmetica, 16(4), 413–424 (1970)

    Article  MathSciNet  Google Scholar 

  8. Eskin, A., Mozes, S., Shah, N.: Unipotent flows and counting lattice points on homogeneous varieties. Annals of Mathematics, 143(2), 253–299 (1996)

    Article  MathSciNet  Google Scholar 

  9. Kleinbock, D. Y., Margulis, G. A.: Flows on homogeneous spaces and Diophantine approximation on manifolds. Annals of Mathematics, 148(2), 339–360 (1998)

    Article  MathSciNet  Google Scholar 

  10. Kleinbock, D., Margulis, G., Wang, J. B.: Metric Diophantine approximation for systems of linear forms via dynamics. International Journal of Number Theory, 6(5), 1139–1168 (2010)

    Article  MathSciNet  Google Scholar 

  11. Kleinbock, D., Weiss, B.: Dirichlet’s theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2(1), 43–62 (2008)

    Article  MathSciNet  Google Scholar 

  12. Mozes, S., Shah, N.: On the space of ergodic invariant measures of unipotent flows. Ergodic Theory and Dynamical Systems, 15(1), 149–159 (1995)

    Article  MathSciNet  Google Scholar 

  13. Ratner, M.: On Raghunathan’s measure conjecture. Annals of Mathematics, 134(2), 545–607 (1991)

    Article  MathSciNet  Google Scholar 

  14. Ratner, M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J., 63(1), 235–280 (1991).

    Article  MathSciNet  Google Scholar 

  15. Shah, N. A.: Equidistribution of expanding translates of curves and Dirichlet’s theorem on Diophantine approximation. Inventiones Mathematicae, 177(3), 509–532 (2009)

    Article  MathSciNet  Google Scholar 

  16. Shah, N. A.: Limiting distributions of curves under geodesic flow on hyperbolic manifolds. Duke Mathematical Journal, 148(2), 251–279 (2009)

    MathSciNet  MATH  Google Scholar 

  17. Shah, N. A.: Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds. Duke Math. J., 148(2), 281–304 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Shah, N. A.: Expanding translates of curves and Dirichlet—Minkowski theorem on linear forms. J. Amer. Math. Soc., 23(2), 563–589 (2010)

    Article  MathSciNet  Google Scholar 

  19. Shah, N. A.: Limit distributions of expanding translates of certain orbits on homogeneous spaces. Proc. Indian Acad. Sci. Math. Sci., 106(2), 105–125 (1996)

    Article  MathSciNet  Google Scholar 

  20. Shah, N. A., Yang, L.: Equidistribution of curves in homogeneous spaces and Dirichlet’s approximation theorem for matrices. Discrete Contin. Dyn. Syst. A, 40(9), 5247–5287 (2020)

    Article  MathSciNet  Google Scholar 

  21. Yang, L.: Expanding curves in T1(ℍn) under geodesic flow and equidistribution in homogeneous spaces. Israel J. Math., 216(1), 389–413 (2016)

    Article  MathSciNet  Google Scholar 

  22. Yang, L.: Equidistribution of expanding curves in homogeneous spaces and Diophantine approximation for square matrices. Proc. Amer. Math. Soc., 144(12), 5291–5308 (2016)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author thanks Nimish Shah for suggesting this problem to him and the anonymous referee for valuable suggestions.

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Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610000, P. R. China

    Lei Yang

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  1. Lei Yang
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Correspondence to Lei Yang.

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Supported by NSFC (Grant No. 11801384) and the Fundamental Research Funds for the Central Universities (Grant No. YJ201769)

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Yang, L. Equidistribution of Expanding Translates of Curves in Homogeneous Spaces with the Action of (SO(n, 1)k. Acta. Math. Sin.-English Ser. 38, 205–224 (2022). https://doi.org/10.1007/s10114-022-0459-1

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  • DOI: https://doi.org/10.1007/s10114-022-0459-1

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