The Bohnenblust–Hille inequality for multilinear forms, published in Annals
of Mathematics in 1931, is now known to be part of a broad family of
anisotropic inequalities that also includes, for example, the mixed Littlewood
and Hardy–Littlewood inequalities. In the real scalar case, the asymptotic
growth of the constants appearing in these inequalities plays a crucial role
in several applications across fields such as Quantum Information Theory and
Computer Science. In this paper, we classify the exponents associated with
these inequalities into distinct classes via an equivalence relation, and we
provide a complete description of the growth behavior of the constants in each
class. In particular, we show that the constants of the anisotropic m-linear
Bohnenblust–Hille inequalities over the real scalars have sub-exponential
growth if, and only if, the sequence of associated exponents is equivalent to
the classical exponents \(\left(\frac{2m}{m+1},\ldots,\frac{2m}{m+1}\right)_{m=1}^{∞}\) thereby solving a problem proposed by the third-named author and E. Teixeira in 2018.
@article{MR4989780,author={Costa Jr., Fernando and N\'u\~nez-Alarc\'on, Daniel and Pellegrino, Daniel and Raposo, Jr., Anselmo},title={<a href="https://doi.org/10.1016/j.jmaa.2025.130220">Characterizing the growth of the anisotropic
{B}ohnenblust-{H}ille constants</a>},journal={J. Math. Anal. Appl.},fjournal={Journal of Mathematical Analysis and Applications},volume={556},year={2026},number={1},pages={Paper No. 130220, 18},issn={0022-247X,1096-0813},mrclass={46G25 (46B45 47H60)},mrnumber={4989780},doi={10.1016/j.jmaa.2025.130220},url={https://doi.org/10.1016/j.jmaa.2025.130220},}
We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When \(A\) is a rooted directed tree, we show that the set of hypercyclic vectors of any weighted backward shift operator on the space \(c_0(A)\) or \(\ell^1(A)\) is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on \(\ell^p(A), 1<p<+∞\). Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing weighted backward shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.
@article{hcalgtrees,author={Abbar, Arafat and Costa Jr., Fernando},title={<a href="https://doi.org/10.1007/s13398-026-01871-z">Hypercyclic algebras for weighted shifts on trees</a>},journal={Rev. Real Acad. Cienc. Exactas Fis. Nat. - A: Mat.},fjournal={Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas},volume={120},year={2026},number={3},pages={76},issn={1579-1505},doi={10.1007/s13398-026-01871-z},url={https://doi.org/10.1007/s13398-026-01871-z},}
We prove that every separable infinite-dimensional Fréchet algebra \(X\)admits a continuous linear operator \(T\)supporting a dense invariant hypercyclic algebra, giving an affirmative answer to a question of Bayart, Costa Jr. and Papathanasiou. In fact, every dense countable-dimensional subalgebra \(\mathcal A\)of \(X\)can be prescribed as an invariant hypercyclic algebra. When \(X\)admits a continuous norm, the operator can additionally be chosen in the form \(T=I+K\), where \(K\)is nuclear, so that \(\mathcal A = \operatorname{span}\,\operatorname{Orb}(a,T)\).
@article{Costa2026,author={Costa Jr., Fernando and Rocha, {\'A}lvaro},title={<a href="https://doi.org/10.1007/s00574-026-00534-2">Existence of Hypercyclic Algebras in {Fr{\'e}chet} Algebras</a>},journal={Bull. Braz. Math. Soc. (N.S.)},fjournal={Bulletin of the Brazilian Mathematical Society, New Series},volume={57},year={2026},month=oct,eid={43},number={4},pages={43},issn={1678-7714},doi={10.1007/s00574-026-00534-2},url={https://doi.org/10.1007/s00574-026-00534-2},}
Given a family \(\mathcal T = (T_λ)_λ∈Λ\)of bounded linear operators acting on the same \(F\)-space \(X\)and indexed by a set of parameters \(Λ⊂\mathbb R^d\), \(d\geq1\), we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of \(\mathcal T\), one of the following three dynamical properties: hypercyclicity, upper frequent hypercyclicity, and frequent hypercyclicity. Our approach relies on associating to \(\mathcal T\)a "Lipschitz constant function" \(F\), whose growth encodes the interaction between the operators and the parameter set \(Λ\). We show that the asymptotic behavior of \(F\)determines sharp thresholds between existence and non-existence of common vectors. In the hypercyclicity setting, we identify the divergence of \(∑1/F(n)^s\), where \(s\)is the Hausdorff dimension of \(Λ\), as a natural condition for existence, and prove complementary non-existence results, which are optimal for weighted shifts. For upper frequent hypercyclicity, we obtain criteria of non-existence in terms of series of the form \(∑1/F(γ^n)\)for parameter sets in \(d\)-dimensions. Considered in the one-dimensional case, these criteria are optimal and allow us to contrast known existence results of Mestiri for logarithmic-type growth with non-existence beyond this scale. For frequent hypercyclicity, we establish a dichotomy showing that common vectors exist essentially only when \(F\)is bounded, while any unbounded growth prevents their existence. Our results apply to a broad class of parameter sets, including self-similar fractals, providing a unified perspective on how growth conditions and geometric features of \(Λ\)determine common dynamical behavior. Finally, in the specific context of weighted shifts, we prove that two operators do not necessarily share the same frequently hypercyclic vectors if the sequence of their weight products ratios admits two distinct non-zero cluster points, establishing the optimality of a result of Grivaux, Matheron and Menet.
@misc{CEMM26common,author={Costa Jr., Fernando and Rocha, {\'A}lvaro and Ernst, Romuald and Mestiri, Monia and Mouze, Augustin},title={<a href="https://arxiv.org/abs/2609.19219">Existence and non-existence in common hypercyclicity and its frequent variants</a>},year={2026},eprint={2609.19219},url={https://arxiv.org/abs/2609.19219},}
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},}
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},}
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},}
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},}
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},}
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: 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},}