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 \(\Bbb R^d\), for all \(d ≥1\).
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.
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 \(λ∈Λ\mapsto T_λ^nx\), \(x∈X\), \(n∈\mathbb N\).
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 \(w_n\)\(_(\)\(_λ\)\(_)=1+λ/n\).
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.
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 \(L(^3\Bbb R^3)\).