Thèmes de Recherche

Systèmes dynamiques et Théorie ergodique Théorie métrique des nombres Approximation diophantienne Multifractals Probabilités Analyse harmonique

Thèse

Études sur la Récurrence de Certains Systèmes Dynamiques Topologiques et Arithmétiques, 198 pages.
Soutenue : le 20 mai 2008 Très Honorable,
Directeur : Ai-Hua FAN; Président: Jean-Pierre KAHANE; Rapporteurs: Yann BUGEAUD, Jörg SCHMELING; Examinateurs: Jean-Luc CHABERT, Fabien DURAND, Bernard HOST.

Habilitation à Diriger des Recherches

Différents aspects de systèmes dynamiques réels et $p$-adiques,
Soutenue : le 12 décembre 2017.
Président : Pierre ARNOUX; Rapporteurs : Julien BARRAL, Charles FAVRE, Mark POLLICOTT; Examinateurs : Valérie BERTHÉ, Yann BUGEAUD, Stéphane SEURET.

Publications

(A summary of my research works can be find here.)

Analyse multifractale des systèmes dynamiques, théorie métrique des nombres, approximation diophantienne (mot clé : dimension de Hausdorff):

  1. Generic points in systems of specification and Banach valued Birkhoff ergodic average, Discrete Contin. Dyn. Syst., 21 (2008) 1103-1128, (with Ai-Hua Fan and Jacques Peyrière) arXiv
  2. Dimension of some non-normal continued fraction sets, Math. Proc. Camb. Phil. Soc., 145 (2008), no. 1, 215-225, (with Ji-Hua Ma and Bao-Wei Wang) PDF
  3. On Khintchine exponents and Lyapunov exponents of continued fractions, Ergodic Theory and Dynamical Systems, 29 (2009), 73-109, (with Ai-Hua Fan, Bao-Wei Wang and Jun Wu) arXiv
  4. On the frequency of partial quotients of regular continued fractions, Math. Proc. Camb. Phil. Soc., 148 (2010), 179-192, (with Ai-Hua Fan and Ji-Hua Ma) arXiv
  5. Dimension of Besicovitch-Eggleston sets in the countable symbolic space, Nonlinearity, 23 (2010) 1185-1197, (with Ai-Hua Fan, Ji-Hua Ma and Bao-Wei Wang) PDF
  6. Level sets of multiple ergodic averages, Monatshefte für Mathematik, 168, no. 1, (2012), 17-26, (with Ai-Hua Fan and Ji-Hua Ma) arXiv
  7. Diophantine approximation by orbits of expanding Markov maps, Ergodic Theory and Dynamical Systems, 33, no. 2, (2013), 585-608, (with Stéphane Seuret) arXiv
  8. Inhomogeneous Diophantine approximation with general error functions, Acta Arith., 160 (2013), 25-35, (with Michał Rams) arXiv
  9. Multifractal analysis of some multiple ergodic averages for the systems with non-constant Lyapunov exponents, Real Analysis Exchange, 39, no. 1(2013), 1-14, (with Michał Rams) arXiv
  10. On the fast Khintchine spectrum in continued fractions, Monatshefte für Mathematik, 171(2013), 329-340, (with Ai-Hua Fan, Bao-Wei Wang and Jun Wu) arXiv
  11. Multifractal analysis for expanding interval maps with infinitely many branches, Trans. Amer. Math. Soc. 367 (2015), 1847-1870, (with Ai-Hua Fan, Thomas Jordan, and Michał Rams) arXiv
  12. Uniform Diophantine approximation related to $b$-ary and $\beta$-expansions, Ergodic Theory and Dynamical Systems, 36, no. 1, (2016), 1-22, (with Yann Bugeaud) arXiv
  13. Subexponentially increasing sum of partial quotients in continued fraction expansions, Math. Proc. Camb. Phil. Soc., 160 (2016), 401-412, (with Michał Rams) arXiv
  14. Upper and lower fast Khintchine spectra in continued fractions, Monatshefte für Mathematik, 180 (2016), 65-81, (with Michał Rams) arXiv
  15. Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential, Fractals, Vol. 26, No. 3 (2018) 1850026 (13 pages), (with Dong Han Kim, Michał Rams and Bao-Wei Wang) arXiv
  16. Multifractal analysis of some multiple ergodic averages in linear Cookie-Cutter dynamical systems, Mathematische Zeitschrif, (2018) 290:63-81, (with Ai-Hua Fan and Meng Wu) arXiv
  17. On the shortest distance between orbits and the longest common substring problem, Advances in Mathematics, 344, (2019), 311-339, (with Vanessa Barros, Jérôme Rousseau) arXiv
  18. Metrical results on the distribution of fractional parts of powers of real numbers, Proceedings of Edinburgh Mathematical Society, 62 (2019), 505-521, (with Michał Rams and Yann Bugeaud) arXiv
  19. Dirichlet uniformly well-approximated numbers, International Mathematics Research Notices, 24 (2019), 7691–7732, (with Dong Han Kim) arXiv
  20. Hausdorff dimension of weighted singular vectors in R^2, Journal of the European Mathematical Society, 22 (2020), 833–875, (with Ronggang Shi, Omri Solan and Nattalie Tamam) arXiv
  21. Normal sequences with given limits of multiple ergodic averages, Publicacions Matemàtiques, 65(1), (2021), 271 - 290, (with Michał Rams) arXiv
  22. Big Birkhoff sums in d-decaying Gauss like iterated function systems, accepted by Studia Mathematica, (with Michał Rams) arXiv
  23. Uniform random covering problems, accepted by International Mathematics Research Notices, (with Henna Koivusalo and Tomas Persson) arXiv

Systèmes dynamiques p-adiques, mesures de Gibbs p-adiques, conjecture de Fuglede p-adique (mot clé : p-adique) :

  1. p-adic repellers in Qp are subshifts of finite type, C. R. Acad. Sci. Paris, Ser. I 344 (2007) 219-224, (with Ai-Hua Fan, Yue-Fei Wang and Dan Zhou)
  2. On minimal decomposition of p-adic polynomial dynamical systems, Advances in Mathematics, 228, no. 4, (2011), 2116-2144, (with Ai-Hua Fan) arXiv
  3. On minimal decomposition of p-adic homographic dynamical systems, Advances in Mathematics, 257 (2014), 92-135, (with Ai-Hua Fan, Shi-Lei Fan and Yue-Fei Wang) arXiv
  4. Dynamics of convergent power series on the integral ring of a finite extension of Q_p, Journal of Differential Equations, 259 (2015)1628-1648, (with Shi-Lei Fan) arXiv
  5. Dynamics of the square mapping on the ring of p-adic integers, Proc. Amer. Math. Soc., 144 (2016), 1183-1196, (with Shi-Lei Fan) arXiv
  6. Dynamical structures of Chebyshev polynomials on Z_2, Journal of Number Theory, 169 (2016), 174-182, (with Shi-Lei Fan) arXiv
  7. Minimality of p-adic rational maps with good reduction, Discrete and Continuous Dynamical Systems -A, 37 (2017), 3161-3182, (with Ai-Hua Fan, Shi-Lei Fan and Yue-Fei Wang) arXiv
  8. Rational map ax+1/x on the projective line over Q_p, Contemporary Mathematics: Proccedings of 14th International Conference on P-adic Functional Analysis, 704(2018), 217-230, (with Shi-Lei Fan) arXiv
  9. Rational map ax+1/x on the projective line over Q_2, Sci China Math, 2018, 61: 2221-2236, (with Shi-Lei Fan) arXiv
  10. Periodic p-adic Gibbs measures of q-states Potts model on Cayley tree: The chaos implies the vastness of p-adic Gibbs measures, Journal of Statistical Physics, 171(2018),1000-1034, (with Mohd Ali Khameini Ahmad and Mansoor Saburov) arXiv
  11. Fuglede's Conjecture holds in Qp, Math. Ann., 375 (2019), 315–341, (with Ai-Hua Fan, Shi-Lei Fan and Ruxi Shi) arXiv
  12. p-adic rational dynamical systems, SCIENTIA SINICA Mathematica, in Chinese, 中国科学 : 数学 2019 年 第 49 卷 第 11 期, (with Ai-Hua Fan, Shi-Lei Fan and Yue-Fei Wang)
  13. p-adic boundary laws and Markov chains on trees, Letters in Mathematical Physics, 110(10), (2020), 2725-2741, (with Arnaud Le Ny, Utkir Rozikov) arXiv

Systèmes dynamiques et numération :

  1. Dynamical properties of negative beta transformation, Ergodic Theory and Dynamical Systems, 32, no. 5, (2012), 1673-1690, (with Wolfgang Steiner) arXiv
  2. Odd-odd continued fraction algorithm, accepted by Monatshefte für Mathematik, (with Dong Han Kim, Seul Bee LeearXiv
  3. Stationary determinantal processes: ψ-mixing property and Lq-dimensions, accepted by Stochastic Processes and their Applications (with Shi-Lei Fan, Yanqi Qiu) arXiv

Articles soumis et prepublications

Articles en préparation

Projets en cours


Slides

Séjours de Recheche

Liste de visites plus que 7 jours

Encadrement de Thèse

Thèses soutenues :

  1. Ruxi SHI, Étude sur la conjecture de Fuglede et les suites oscillantes,
    ancien  élève normalien (Ulm), Contrat doctoral du Ministère de l'Education Nationale
    en cotutelle avec Ai-Hua FAN (Université de Picardie Jules Verne), soutenue le 26/06/2018.
    Postdoc à Université d'Oulu, puis à Institute of Mathematics, Polish Academy of Sciences.

  2. Ali AHMAD, Mesures de Gibbs p-adiques sur les arbres de Cayley et systèmes dynamiques sur le corps des nombres p-adiques,
    Bourse de l'Ambassade de France en Malaisie et bourse Labex Bézout, séjours en France du 01/10/2016 au 31/03/2018
    en cotutelle avec Mansoor SABUROV et Chin Hee PAH (Universit ́e Islamique Internationale de Malaisie)
    en cotutelle avec Arnaud LE NY (Université Paris-Est Créteil), soutenue le 29/08/2019.
    Lecturer à Universiti Teknologi Malaysia.

  3. Wanlou WU, Taux de croissance du nombre d'orbites périodiques pour les champs de vecteurs et approximation diophantienne uniforme des bêta-transformations,
    Bourse du China Scholarship Council, séjours en France du 08/10/2017 au 07/10/2019
    en cotutelle avec Dawei YANG (Université de Soochow, Chine), soutenue le 31/10/2019.
    Professeur associé à Jiangsu Normal University.

  4. Lixuan ZHENG, Bêta-développements et propriétés fractales,
    Bourse du China Scholarship Council, séjours en France du 29/09/2017 au 28/09/2019
    en cotutelle avec Min WU (Université de technologie de Chine méridionale, Chine), soutenue le 18/12/2019.
    Lectrice à Guangdong University of Finance & Economics.

  5. Kunkun SONG, Certaines contributions à l'analyse multifractale en théorie des fractions continues et à la transformée de Fourier des mesures,
    Bourse du China Scholarship Council, séjours en France du 27/09/2018 au 26/09/2020
    en cotutelle avec Jihua MA (Université de Wuhan, Chine), soutenue le 05/12/2020.
    Lecturer à Hunan Normal University.

Thèses en cours :
1. Qian ZHANG, Analyse multifractale multivariée,
    Contrat doctoral du Ministère de l'Education Nationale
    en cotutelle avec Stéphane JAFFARD (Université Paris-Est Créteil, depuis le 20/09/2021.

2. Yubin HE, Quelques études sur les points bien approchés dans des ensembles de type de Cantor
    Bourse du China Scholarship Council, séjours en France du 14/10/2021 au 13/10/2023
    en cotutelle avec Ying XIONG (Université de technologie de Chine méridionale, Chine), depuis le 14/10/2021.

3. Jing FENG, Problèmes fractals dans l'approximation diophantienne uniforme
    Bourse du China Scholarship Council, séjours en France du 18/11/2021 au 17/11/2023
    en cotutelle avec Jian XU (Université des Sciences et Technologie de Huazhong, Chine), depuis le 18/11/2021.

Collaborateurs

Mohd Ali Khameini AHMAD, Arnaud LE NY, Vanessa BARROS, Yann BUGEAUD, Ai-Hua FAN, Shi-Lei FAN, Yubin HE, Thomas JORDAN, Dong Han KIM, Henna KOIVUSALO, Bing LI, Seul Bee LEE, Ji-Hua MA, Tomas PERSSON, Jacques PEYRIERE, Yanqi QIU, Michał RAMS, Utkir ROZIKOV, Jérôme ROUSSEAU, Mansoor SABUROV, Stéphane SEURET, Ronggang SHI, Ruxi SHI, Wolfgang STEINER, Omri SOLAN, Nattalie TAMAM, Sanju VELANI, Bao-Wei WANG, Yue-Fei WANG, Meng WU, Jun WU, Dan ZHOU, Evgeniy ZORIN.

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Last updated: August, 2022