Multiple recurrence and hypercyclicity


  • Rodrigo Cardeccia
  • Santiago Muro



We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where $\mathcal F$ is the family of subsets of the natural numbers containing arbitrarily long arithmetic progressions. We prove several properties of hypercyclic multiply recurrent operators, we characterize those operators which are weakly mixing and multiply recurrent, and we show that there are operators that are multiply recurrent and hypercyclic but not weakly mixing.


Ansari, S. I., Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390.

Badea, C., and Grivaux, S., Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766–793.

Bayart, F., and Grivaux, S., Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), no. 11, 5083–5117.

Bayart, F., and Matheron, E., Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces, J. Funct. Anal. 250 (2007), no. 2, 426–441.

Bayart, F., and Matheron, E., Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009.

Bayart, F. and Matheron, E., (Non-)weakly mixing operators and hypercyclicity sets, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 1, 1–35.

Bernal-González, L., On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003–1010.

Bernal-González, L., Disjoint hypercyclic operators, Studia Math. 182 (2007), no. 2, 113–131.

Bès, J., Martin, O., Peris, A., and Shkarin, S., Disjoint mixing operators, J. Funct. Anal. 263 (2012), no. 5, 1283–1322.

Bès, J., Menet, Q., Peris, A., and Puig, Y., Recurrence properties of hypercyclic operators, Math. Ann. 366 (2016), no. 1-2, 545–572.

Bès, J., Menet, Q., Peris, A., and Puig, Y., Strong transitivity properties for operators, J. Differential Equations 266 (2019), no. 2-3, 1313–1337.

Bès, J. and Peris, A., Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94–112.

Bès, J. and Peris, A., Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007), no. 1, 297–315.

Bonet, J., and Peris, A., Hypercyclic operators on non-normable Fréchet spaces., J. Funct. Anal. 159 (1998), no. 2, 587–595.

Bonilla, A., and Grosse-Erdmann, K.-G., Upper frequent hypercyclicity and related notions, Rev. Mat. Complut. 31 (2018), no. 3, 673–711.

Bonilla, A., Grosse-Erdmann, K.-G., López-Martínez, A., and Peris, A., Frequently recurrent operators, arXiv:2006.11428

Cardeccia, R., Dynamics of homogeneous mappings, Ph.D. thesis, Universidad de Buenos Aires, 2020.

Cardeccia, R., and Muro, S., Arithmetic progressions and chaos in linear dynamics, Integral Equations Operator Theory 94 (2022), no. 2, no. 11.

Cardeccia, R., and Muro, S., Frequently recurrence properties and block families, arXiv:2204.13542.

Chen, C.-C., Recurrence for weighted translations on groups, Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 2, 443–452.

Chen, C.-C., Recurrence of cosine operator functions on groups, Canad. Math. Bull. 59 (2016), no. 4, 693–704.

Chen, C.-C., Dynamics of weighted translations on Orlicz spaces, Collect. Math. 71 (2020), no. 1, 173–187.

Costakis, G., Manoussos, A., and Parissis, I., Recurrent linear operators, Complex Anal. Oper. Theory 8 (2014), no. 8, 1601–1643.

Costakis, G. and Parissis, I., Szemerédi's theorem, frequent hypercyclicity and multiple recurrence, Math. Scand. 110 (2012), no. 2, 251–272.

De La Rosa, M., and Read, C., A hypercyclic operator whose direct sum $Toplus T$ is not hypercyclic, J. Operator Theory 61 (2009), no. 2, 369–380.

Ernst, R., Esser, C., and Menet, Q., $mathcal U$-frequent hypercyclicity notions and related weighted densities, Israel J. Math. 241 (2021), no. 2, 817–848.

Furstenberg, H., Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981.

Gowers, W. T., A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588.

Green, B. and Tao, T., The primes contain arbitrarily long arithmetic progressions, Ann. Math. (2) 167 (2008), no. 2, 481–547.

Grosse-Erdmann, K.-G., and Peris Manguillot, A., Linear chaos, Universitext. Springer, London, 2011.

Kwietniak, D., Li, J., Oprocha, P., and Ye, X., Multi-recurrence and van der Waerden systems, Sci. China Math. 60 (2017), no. 1, 59–82.

Puig, Y., Frequent hypercyclicity and piecewise syndetic recurrence sets, arXiv:1703.09172.

Puig, Y., A mixing operator $T$ for which $(T,T^2)$ is not disjoint transitive, Studia Math. 237 (2017), no. 3, 283–296.

Puig de Dios, Y., Linear dynamics and recurrence properties defined via essential idempotents of $beta mathbb N$, Ergodic Theory Dynam. Systems 38 (2018), no. 1, 285–300.

Salas, H. N., Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993–1004.

Shkarin, S., On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc. 137 (2009), no. 1, 123–134.

Szemerédi, E., On sets of integers containing no $k$ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.



How to Cite

Cardeccia, R., & Muro, S. (2022). Multiple recurrence and hypercyclicity. MATHEMATICA SCANDINAVICA, 128(3).