I am a CNRS researcher in the Laurent Schwartz center for Mathematics, École Polytechnique. I am interested in dynamics on Lie groups.
Email: jialun.li AT polytechnique.edu
Publications and preprints
[14] On the Dimension of Limit Sets on RP^2 via Stationary Measures II: variational principles and applications, with Yuxiang Jiao, Wenyu Pan, Disheng Xu, arXiv:2311.10262
[13] On the dimension of limit sets on RP^2 via stationary measures I: the theory and Applications, with Wenyu Pan, Disheng Xu, arXiv:2311.10265
[12] Exponential mixing of frame flows for geometrically finite hyperbolic manifolds, with Pratyush Sarkar, Wenyu Pan, arXiv:2302.03798
[11] Stationary measures for SL2(ℝ)-actions on homogeneous bundles over flag varieties, with Alex Gorodnik, Cagri Sert, arXiv:2211.06911
[10] Equidistribution and counting of periodic tori in the space of Weyl chambers, with Nguyen-Thi Dang, accepted by Commentarii Mathematici Helvetici,
See arXiv:2202.08323 for an older version containing the case of SL(n,Z).[9] Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps, with Wenyu Pan, Invent. math. 231, 931–1021 (2023).
[8] Appendix of: The space of homogeneous probability measures on Γ∖X is compact, by Christopher Daw, Alexander Gorodnik, Emmanuel Ullmo, Math. Ann. 386, 987–1016 (2023).
[7] Fourier transform of self-affine measures, with Tuomas Sahlsten, Advances in Mathematics 374 (2020).
[6] Trigonometric Series and Self-similar Sets, with Tuomas Sahlsten, J. Eur. Math. Soc. 24 (2022), no. 1, pp. 341–368
[1], [6] and [7] use a similar idea, renewal theorem implies decay of Fourier transform of measures on fractal sets. [6] is the simplest non trivial case and is the easiest to read.[5] Kleinian Schottky groups, Patterson-Sullivan measures, and Fourier decay, with an appendix on stationarity of Patterson-Sullivan measures, with Frédéric Naud and Wenyu Pan, Duke Math. J. 170, issue 4, (2021) pp. 775 - 825.
[4] Fourier decay, Renewal theorem and Spectral gaps for random walks on split semisimple Lie groups, Annales Scientifiques de l'ÉNS, Tome 55, Fasc.6, pp 1613-1686, 2022
[4] and [5] generalize the idea of Bourgain-Dyatlov, non-concentration and discretized sum-product estimates imply decay of Fourier transform of Furstenberg measures. A draft on SL_2(R) maybe helpful to understand the method.See also a post of Carlos Matheus for my talk on this topic.[3] Discretized Sum-product and Fourier decay in Rn, Journal d'Analyse Mathématique, 143, (2021) pp. 763–800.
[2] Finiteness of Small Eigenvalues of Geometrically Finite Rank one Locally Symmetric Manifolds, Mathematical Research Letters, 27, number 2 (2020) pp. 465 – 500.
[1] Decrease of Fourier Coefficients of Stationary Measures, Mathematische Annalen, 372, (2018) pp. 1189–1238.
Short CV
2019-2023 Postdoc in University of Zurich
2015-2018 Ph.D. of Mathematics in University of Bordeaux
Thesis with the title “Harmonic analysis of Stationary measures” under the direction of Jean-Francois Quint defended 4 December 2018, at l’Institut de Mathématiques de Bordeaux.
2013-2016 Student in Ecole Normale Supérieure Paris
2010-2014 License of mathematics, in Tsinghua University, Beijing, China
Organization:
Diagonal actions in the space of lattices, Palaiseau, 4-6 March 2024