Joint transitivity for linear iterates
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e34

Voir la notice de l'article provenant de la source Cambridge University Press

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems.
Donoso, Sebastián; Koutsogiannis, Andreas; Sun, Wenbo. Joint transitivity for linear iterates. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e34. doi: 10.1017/fms.2024.138
@article{10_1017_fms_2024_138,
     author = {Donoso, Sebasti\'an and Koutsogiannis, Andreas and Sun, Wenbo},
     title = {Joint transitivity for linear iterates},
     journal = {Forum of Mathematics, Sigma},
     pages = {e34},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2024.138},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.138/}
}
TY  - JOUR
AU  - Donoso, Sebastián
AU  - Koutsogiannis, Andreas
AU  - Sun, Wenbo
TI  - Joint transitivity for linear iterates
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e34
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.138/
DO  - 10.1017/fms.2024.138
ID  - 10_1017_fms_2024_138
ER  - 
%0 Journal Article
%A Donoso, Sebastián
%A Koutsogiannis, Andreas
%A Sun, Wenbo
%T Joint transitivity for linear iterates
%J Forum of Mathematics, Sigma
%D 2025
%P e34
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.138/
%R 10.1017/fms.2024.138
%F 10_1017_fms_2024_138

[1] Akin, E. and Glasner, E., ‘Residual properties and almost equicontinuity’, J. Anal. Math. 84 (2001), 243–286. Google Scholar | DOI

[2] Auslander, J., Minimal Flows and Their Extensions (North-Holland Mathematics Studies) vol. 115 (North-Holland Publishing Co., Amsterdam, 1988). Google Scholar

[3] Auslander, J. and Guerin, M., ‘Regional proximality and the prolongation’, Forum Math. 9 (1997), 761–774. Google Scholar | DOI

[4] Berend, D. and Bergelson, V., ‘Jointly ergodic measure-preserving transformations’, Israel J. Math. 49 (1984), 307–314. Google Scholar | DOI

[5] Bergelson, V., ‘Weakly mixing PET’, Ergodic Theory Dynam. Systems 7(1987), 337–349. Google Scholar | DOI

[6] Bergelson, V., Leibman, A. and Son, Y., ‘Joint ergodicity along generalized linear functions’, Ergodic Theory Dynam. Systems 36 (2016), 2044–2075. Google Scholar | DOI

[7] Bergelson, V., Moreira, J. and Richter, F. K., ‘Multiple ergodic averages along functions from a Hardy field: Convergence, recurrence and combinatorial applications’, Adv. Math. 443 (2024), Paper No. 109597. Google Scholar | DOI

[8] Cao, Y. and Shao, S., ‘Topological mild mixing of all orders along polynomials’, Discrete Contin. Dyn. Syst. 42 (2022), 1163–1184. Google Scholar | DOI

[9] Chu, Q., Frantzikinakis, N. and Host, B., ‘Ergodic averages of commuting transformations with distinct degree polynomial iterates’, Proc. Lond. Math. Soc. (3) 102 (2011), 801–842. Google Scholar | DOI

[10] De Vries, J., Elements of Topological Dynamics (Mathematics and Its Applications) vol. 257 (Kluwer Academic Publishers Group, Dordrecht, 1993). Google Scholar | DOI

[11] Donoso, S., Durand, F., Maass, A. and Petite, S., ‘On automorphism groups of low complexity subshifts’, Ergodic Theory Dynam. Systems 36 (2016), 64–95. Google Scholar | DOI

[12] Donoso, S., Ferré Moragues, A., Koutsogiannis, A. and Sun, W., ‘Decomposition of multicorrelation sequences and joint ergodicity’, Ergodic Theory Dynam. Systems 44 (2024), 432–480. Google Scholar | DOI

[13] Donoso, S., Koutsogiannis, A., Kuca, B., Tsinas, K. and Sun, W., ‘Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences’, Preprint, 2024, . Google Scholar | arXiv

[14] Donoso, S., Koutsogiannis, A. and Sun, W., ‘Pointwise multiple averages for sublinear functions’, Ergodic Theory Dynam. Systems 40 (2020), 1594–1618. Google Scholar | DOI

[15] Donoso, S., Koutsogiannis, A. and Sun, W., ‘Seminorms for multiple averages along polynomials and applications to joint ergodicity’, J. Anal. Math. 146 (2022), 1–64. Google Scholar | DOI

[16] Donoso, S., Koutsogiannis, A. and Sun, W., ‘Joint ergodicity for functions of polynomial growth’, to appear in Israel J. Math. Google Scholar

[17] Donoso, S. and Sun, W., ‘Dynamical cubes and a criteria for systems having product extensions’, J. Mod. Dyn. 9 (2015), 365–405. Google Scholar | DOI

[18] Ellis, R. and Gottschalk, W. H., ‘Homomorphisms of transformation groups’, Trans. Amer. Math. Soc. 94 (1960), 258–271. Google Scholar | DOI

[19] Frantzikinakis, N., ‘Multiple recurrence and convergence for Hardy sequences of polynomial growth’, J. Anal. Math. 112 (2010), 79–135. Google Scholar | DOI

[20] Frantzikinakis, N., ‘A multidimensional Szemerédi theorem for Hardy sequences of different growth’, Trans. Amer. Math. Soc. 367 (2015), 5653–5692. Google Scholar | DOI

[21] Frantzikinakis, N., ‘Joint ergodicity of sequences’, Adv. Math. 417 (2023), Paper No. 108918. Google Scholar | DOI

[22] Frantzikinakis, N. and Kra, B., ‘Polynomial averages converge to the product of integrals’, Israel J. Math. 148 (2005), 267–276. Google Scholar | DOI

[23] Frantzikinakis, N. and Kuca, B., ‘Joint ergodicity for commuting transformations and applications to polynomial sequences’, Invent. Math. 239 (2025), 621–706. Google Scholar | DOI

[24] Frantzikinakis, N. and Kuca, B., ‘Seminorm control for ergodic averages with commuting transformations and pairwise dependent polynomial iterates’, Ergodic Theory Dynam. Systems 43 (2023), 4074–4137. Google Scholar | DOI

[25] Furstenberg, H., ‘Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions’, J. Anal. Math. 31 (1977), 204–256. Google Scholar | DOI

[26] Glasner, E., ‘Topological ergodic decompositions and applications to products of powers of a minimal transformation’, J. Anal. Math. 64 (1994), 241–262. Google Scholar | DOI

[27] Host, B., Kra, B. and Maass, A., ‘Nilsequences and a structure theorem for topological dynamical systems’, Adv. Math. 224 (2010), 103–129. Google Scholar | DOI

[28] Huang, W., Shao, S. and Ye, X., ‘Nil Bohr-sets and almost automorphy of higher order’, Mem. Amer. Math. Soc. 241 (2016), v+83. Google Scholar

[29] Huang, W., Shao, S. and Ye, X., ‘Topological correspondence of multiple ergodic averages of nilpotent group actions’, J. Anal. Math. 138 (2019), 687–715. Google Scholar | DOI

[30] Karageorgos, D. and Koutsogiannis, A., ‘Integer part independent polynomial averages and applications along primes’, Studia Math. 249 (2019), 233–257. Google Scholar | DOI

[31] Koutsogiannis, A., ‘Integer part polynomial correlation sequences’, Ergodic Theory Dynam. Systems 38 (2018), 1525–1542. Google Scholar | DOI

[32] Koutsogiannis, A., ‘Multiple ergodic averages for variable polynomials’, Discrete Contin. Dyn. Syst. 42 (2022), 4637–4668. Google Scholar | DOI

[33] Koutsogiannis, A. and Sun, W., ‘Total joint ergodicity for totally ergodic systems’, Preprint, 2023, . Google Scholar | arXiv

[34] Kwietniak, D. and Oprocha, P., ‘On weak mixing, minimality and weak disjointness of all iterates’, Ergodic Theory Dynam. Systems 32 (2012), 1661–1672. Google Scholar | DOI

[35] Lehrer, E., ‘Topological mixing and uniquely ergodic systems’, Israel J. Math. 57 (1987), 239–255. Google Scholar | DOI

[36] Moothathu, T. K. S., ‘Diagonal points having dense orbit’, Colloq. Math. 120 (2010), 127–138. Google Scholar | DOI

[37] Qiu, J., ‘Polynomial orbits in totally minimal systems’, Adv. Math. 432 (2023), Paper No. 109260. Google Scholar | DOI

[38] Shao, S. and Ye, X., ‘Regionally proximal relation of order is an equivalence one for minimal systems and a combinatorial consequence’, Adv. Math. 231 (2012), 1786–1817. Google Scholar | DOI

[39] Tsinas, K., ‘Joint ergodicity of Hardy field sequences’, Trans. Amer. Math. Soc. 376 (2023), 3191–3263. Google Scholar | DOI

[40] Veech, W. A., ‘The equicontinuous structure relation for minimal Abelian transformation groups’, Amer. J. Math. 90 (1968), 723–732. Google Scholar | DOI

[41] Zhang, R. F. and Zhao, J. J., ‘Topological multiple recurrence of weakly mixing minimal systems for generalized polynomials’, Acta Math. Sin. (Engl. Ser.) 37 (2021), 1847–1874. Google Scholar | DOI

Cité par Sources :