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Augustin Mouze

Publications and source records attributed to Augustin Mouze.

8 recordsLinked to original sources
arXiv2026

Existence and non-existence in common hypercyclicity and its frequent variants

Given a family $\mathcal T = (T_λ)_{λ\inΛ}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $Λ\subset \mathbb{R}^d$, $d\geq1$, 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 $\sum 1/F(n)^s$, 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 $\sum 1/F(γ^n)$ 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.

math.DS↗
arXiv2026

Common frequently hypercyclic random vectors

We study common frequently hypercyclic vectors for countable families of weighted backward shifts acting on $\ell_p$ spaces, $1\leq p<\infty$. Using probabilistic techniques, we develop a general existence criterion, complemented by a non-existence result. These insights are then applied to the specific setting of countable families of polynomials of weighted backward shifts, providing conditions under which they share a common frequently hypercyclic vector.

math.FA↗
arXiv2025

Optimal growth of upper frequently hypercyclic functions for some weighted Taylor shifts

We are interested in the optimal growth in terms of $L^p$-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic function on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between the $\mathcal{U}$-frequent hypercyclicity and the hypercyclicity.

math.CA↗
arXiv2021

Universal sequences of composition operators

Let $G$ and $Ω$ be two planar domains. We give necessary and sufficient conditions on a sequence $(ϕ_n)$ of eventually injective holomorphic mappings from $G$ to $Ω$ for the existence of a function $f\in H(Ω)$ whose orbit under the composition by $(ϕ_n)$ is dense in $H(G)$. This extends a result of the same nature obtained by Grosse-Erdmann and Mortini when $G=Ω$. An interconnexion between the topological properties of $G$ and $Ω$ appears. Further, in order to exhibit in a natural way holomorphic functions with wild boundary behaviour on planar domains, we study a certain type of universality for sequences of continuous mappings from a union of Jordan curves to a domain.

math.CV↗
arXiv2021

Common frequent hypercyclicity

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset $Λ$ of the complex plane for the family {$λ$T : $λ$ $\in$ $Λ$} to have a common frequently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclic weighted shifts which do not possess common frequent hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common $α$-frequent hypercyclicity may be vacuous, where the notion of $α$-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is $α$-frequently universal for any sequence $α$ satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is $α$-frequently universal for T , with respect to all such $α$.

math.FA↗
arXiv2020

Universal Taylor series with respect to a prescribed subsequence

For a holomorphic function $f$ in the open unit disc $\mathbb{D}$ and $ζ\in\mathbb{D}$, $S_n(f,ζ)$ denotes the $n$-th partial sum of the Taylor development of $f$ at $ζ$. Given an increasing sequence of positive integers $μ=(μ_n)$, we consider the classes $\mathcal{U}(\mathbb{D},ζ)$ (resp. $\mathcal{U}^{(μ)}(\mathbb{D},ζ)$) of such functions $f$ such that the partial sums $\{S_n(f,ζ):n=1,2,\dots\}$ (resp. $\{S_{μ_n}(f,ζ):n=1,2,\dots\}$) approximate all polynomials uniformly on the compact sets $K\subset\{z\in\mathbb{C}:\vert z\vert\geq 1\}$ with connected complement. We show that these two classes of universal Taylor series coincide if and only if $\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$. In the same spirit, we prove that, for $ζ\ne 0,$ we have the equality $\mathcal{U}^{(μ)}(\mathbb{D},ζ)=\mathcal{U}^{(μ)}(\mathbb{D},0)$ if and only if $\limsup_n\left(\frac{μ_{n+1}}{μ_n}\right)<+\infty$. Finally we deal with the case of real universal Taylor series.

math.CA↗
arXiv2014

Generalized universal series

We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form $\sum_{k=0}^n a_k x_{n,k}$ for given sequences of vectors $(x_{n,k})_{n\geq k\geq 0}$ in a topological vector space X. The algebraic and topological genericity as well as the spaceability are discussed. Then we provide various examples of such generalized universal series which do not proceed from the classical theory. In particular, we build universal series involving Bernstein's polynomials, we obtain a universal series version of MacLane's Theorem, and we extend a result of Tsirivas concerning universal Taylor series on simply connected domains, exploiting Bernstein- Walsh quantitative approximation theorem.

math.FA↗