Publications

2024

1. J. Funct. Anal.

   Self-similar fractals and common hypercyclicity

   Fernando Costa Jr.

   J. Funct. Anal., 2024

   Abs Bib

   We obtain a multi-dimensional generalization of the Costakis-Sambarino criterion for common hypercyclic vectors with optimal consequences on a large class of fractals. Applications include families of products of backward shifts parameterized by any Hölder continuous curve in \(\mathbb R^d\), for all \(d≥1\).

   @article{costa2023self,
     author = {Costa Jr., Fernando},
     title = {<a href="https://doi.org/10.1016/j.jfa.2024.110473">Self-similar fractals and common hypercyclicity</a>},
     journal = {J. Funct. Anal.},
     fjournal = {Journal of Functional Analysis},
     year = {2024},
     pages = {110473},
     issn = {0022-1236},
     mrclass = {47A16(47B37), 28A80, 28A78},
     doi = {10.1016/j.jfa.2024.110473},
     url = {https://doi.org/10.1016/j.jfa.2024.110473},
     keywords = {common hypercyclicity, Hausdorff dimension, self-similar fractal, Hölder curve},
   }

2. arXiv preprint

   Hypercyclic algebras for weighted shifts on trees

   Arafat Abbar , and Fernando Costa Jr.

   2024

   Abs Bib

   We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When \(V\) is a rooted directed tree, we show the set of hypercyclic vectors of any backward weighted shift operator on the space \(c_0(V)\) or \(\ell^1(V)\) is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on \(\ell^p(V)\), \(1<p<+∞\). Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing backward weighted shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.

   @misc{hcalgtrees,
     author = {Abbar, Arafat and Costa Jr., Fernando},
     title = {<a href="https://arxiv.org/abs/2411.14609">Hypercyclic algebras for weighted shifts on trees</a>},
     year = {2024},
     eprint = {2411.14609},
     archiveprefix = {arXiv},
     primaryclass = {math.FA},
     url = {https://arxiv.org/abs/2411.14609},
   }

2022

1. Israel J. Math.

   Disjoint and common hypercyclic algebras

   Frédéric Bayart , Fernando Costa Jr., and Dimitris Papathanasiou

   Israel J. Math., 2022

   Abs Bib

   Over time the concept of hypercyclicity has been explored in different manners and contexts, gaining new forms and applications. In particular when the spacehas an adjacent structure, we can always look for sets of hypercyclic vectors compatible with that framework. In this paper we deal with hypercyclic operators acting on Fréchet sequence algebras and give criteria for the existence of common and disjoint hypercyclic algebras.

   @article{MR4499027,
     author = {Bayart, Fr\'{e}d\'{e}ric and Costa Jr., Fernando and Papathanasiou, Dimitris},
     title = {<a href="https://doi.org/10.1007/s11856-022-2337-z">Disjoint and common hypercyclic algebras</a>},
     journal = {Israel J. Math.},
     fjournal = {Israel Journal of Mathematics},
     volume = {250},
     year = {2022},
     number = {1},
     pages = {211--264},
     issn = {0021-2172,1565-8511},
     mrclass = {47A16 (47B37)},
     mrnumber = {4499027},
     mrreviewer = {Abdellatif\ Bourhim},
     doi = {10.1007/s11856-022-2337-z},
     url = {https://doi.org/10.1007/s11856-022-2337-z},
   }

2. Indiana Univ. Math. J

   Common hypercyclic vectors and dimension of the parameter set

   Frédéric Bayart , Fernando Costa Jr., and Quentin Menet

   Indiana Univ. Math. J., 2022

   Abs Bib

   We investigate the existence of a common hypercyclic vector for a family \((T_λ)\)\(_λ\)\(_∈\)\(_Λ\) of hypercyclic operators acting on the same Banach space \(X\). We give positive and negative results involving the dimension of \(Λ\) and the regularity of each map \(λ∈Λ\mapsto T_λ^nx\), \(x∈X\), \(n∈\mathbb N\).

   @article{MR4481099,
     author = {Bayart, Fr\'{e}d\'{e}ric and Costa Jr., Fernando and Menet, Quentin},
     title = {<a href="https://doi.org/10.1512/iumj.2022.71.9369">Common hypercyclic vectors and dimension of the parameter set</a>},
     journal = {Indiana Univ. Math. J.},
     fjournal = {Indiana University Mathematics Journal},
     volume = {71},
     year = {2022},
     number = {4},
     pages = {1763--1795},
     issn = {0022-2518,1943-5258},
     mrclass = {47A16},
     mrnumber = {4481099},
     doi = {10.1512/iumj.2022.71.9369},
     url = {https://doi.org/10.1512/iumj.2022.71.9369},
   }

3. J. Math. Anal. Appl.

   Common hypercyclic algebras for families of products of backward shifts

   Fernando Costa Jr.

   J. Math. Anal. Appl., 2022

   Abs Bib

   In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a questionraised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on \(\ell_1(\Bbb N)\) with the convolution product for the family of backward shifts \((B_w\)\(_(\)\(_λ\)\(_)\)\()_λ\)\(_>\)\(_0\) induced by the weights \(w_n\)\(_(\)\(_λ\)\(_)=1+λ/n\).

   @article{MR4333146,
     author = {Costa Jr., Fernando},
     title = {<a href="https://doi.org/10.1016/j.jmaa.2021.125768">Common hypercyclic algebras for families of products of
                   backward shifts</a>},
     journal = {J. Math. Anal. Appl.},
     fjournal = {Journal of Mathematical Analysis and Applications},
     volume = {507},
     year = {2022},
     number = {1},
     pages = {Paper No. 125768, 20},
     issn = {0022-247X,1096-0813},
     mrclass = {47A16 (47B37)},
     mrnumber = {4333146},
     mrreviewer = {Yu-Xia\ Liang},
     doi = {10.1016/j.jmaa.2021.125768},
     url = {https://doi.org/10.1016/j.jmaa.2021.125768},
   }

2021

1. Adv. Math.

   Baire theorem and hypercyclic algebras

   Frédéric Bayart , Fernando Costa Jr., and Dimitris Papathanasiou

   Adv. Math., 2021

   Abs Bib

   The question of whether a hypercyclic operator \(T\) acting on a Fréchet algebra \(X\) admits or not an algebra of hypercyclic vectors (but \(0\)) has been addressed in the recent literature. In this paper we give new criteria and characterisations in the context of convolution operators acting on \(H(\Bbb C)\) and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on \(c_0\) admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.

   @article{MR4178905,
     author = {Bayart, Fr\'{e}d\'{e}ric and Costa Jr., Fernando and Papathanasiou, Dimitris},
     title = {<a href="https://doi.org/10.1016/j.aim.2020.107419">Baire theorem and hypercyclic algebras</a>},
     journal = {Adv. Math.},
     fjournal = {Advances in Mathematics},
     volume = {376},
     year = {2021},
     pages = {Paper No. 107419, 58},
     issn = {0001-8708,1090-2082},
     mrclass = {47A16 (46E10 47B48)},
     mrnumber = {4178905},
     mrreviewer = {Jos\'{e}\ Bonet},
     doi = {10.1016/j.aim.2020.107419},
     url = {https://doi.org/10.1016/j.aim.2020.107419},
   }

2018

1. Numer. F. A. Optim.

   The optimal multilinear Bohnenblust-Hille constants: acomputational solution for the real case

   Fernando Costa Jr.

   Numer. Funct. Anal. Optim., 2018

   Abs Bib

   The Bohnenblust–Hille inequality for \(m\)-linear forms was proven in 1931 as a generalization of the famous \(4/3\)-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as \(m\)grows) is unknown, but significant for applications. A recent result, obtained by Cavalcante, Teixeira and Pellegrino, provides a kind of algorithm, composed by finitely many elementary steps, giving as the final outcome the optimal truncated Bohnenblust–Hille constants of any order. But the procedure of Cavalcante et al. has a fairly large number of calculations and computer assistance cannot be avoided. In this paper we present a computational solution to the problem, using the Wolfram Language. We also use this approach to investigate a conjecture raised by Pellegrino and Teixeira, asserting that \(C_m=2^1\)\(^−\)\(^1\)\(^/\)\(^m\) for all \(m∈\Bbb N\) and to reveal interesting unknown facts about the geometry of the closed unit ball in the space \(L(^3\Bbb R^3)\).

   @article{MR3902552,
     author = {Costa Jr., Fernando},
     title = {<a href="https://doi.org/10.1080/01630563.2018.1489414">The optimal multilinear {B}ohnenblust-{H}ille constants&#58; acomputational solution for the real case</a>},
     journal = {Numer. Funct. Anal. Optim.},
     fjournal = {Numerical Functional Analysis and Optimization. An
                   International Journal},
     volume = {39},
     year = {2018},
     number = {15},
     pages = {1656--1668},
     issn = {0163-0563,1532-2467},
     mrclass = {26D15},
     mrnumber = {3902552},
     doi = {10.1080/01630563.2018.1489414},
     url = {https://doi.org/10.1080/01630563.2018.1489414},
   }