arXiv subjects

Antoni López-Martínez

Publications and source records attributed to Antoni López-Martínez.

13 recordsLinked to original sources

arXiv2026

Expansive operators on Fréchet spaces without continuous norm

We investigate the existence of (metrically) expansive operators on Fréchet spaces without continuous norm. We first solve a problem posed by Bernardes, Caraballo, Darji, Fávaro, and Peris in 2025, by showing that there is no expansive operator on the Fréchet space $ω$ of all sequences. We prove this fact by investigating the existence of orbits remaining cofinitely inside some fixed closed subspace of finite codimension. We then give examples of expansive operators on some other Fréchet spaces without continuous norm.

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arXiv2026

Li-Yorke chaos on fuzzy dynamical systems

Given a dynamical system $(X,f)$ we investigate how several variants of Li-Yorke chaos behave with respect to the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, where $\overline{f}$ is the hyperextension of $f$ acting on the space $\mathcal{K}(X)$ of non-empty compact subsets of $X$, and where $\hat{f}$ denotes the Zadeh extension of $f$ acting on the space $\mathcal{F}(X)$ of normal fuzzy subsets of $X$. We first prove that the main variants of Li-Yorke chaos transfer from $(X,f)$ to $(\mathcal{K}(X),\overline{f})$ and from $(\mathcal{K}(X),\overline{f})$ to $(\mathcal{F}(X),\hat{f})$, but that the converse implications do not hold in general. However, combining the notions of proximality and sensitivity we introduce Cantor-dense Li-Yorke chaos, and we prove that this strengthened variant of chaos does transfer from $(\mathcal{F}(X),\hat{f})$ to $(\mathcal{K}(X),\overline{f})$ under natural assumptions.

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arXiv2026

Topological dynamics for the endograph metric II: Extremely radical properties

Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for some contractive and expansive properties, for chain recurrence, chain transitivity and chain mixing, and for the shadowing property, the endograph metric behaves in an extremely radical way. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes concerning the chain-type notions considered and the shadowing property.

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arXiv2026

Topological dynamics for the endograph metric I: Equivalences with other metrics

Given a dynamical system $(X,f)$ we investigate several topological dynamical properties for its Zadeh extension $(\mathcal{F}(X),\hat{f})$ endowed with the endograph metric $d_{E}$. In particular, we prove that for topological $\mathcal{A}$-transitivity, topological $(\ell,\mathcal{A})$-recurrence, Devaney chaos, and for the specification property, the endograph metric behaves similarly to the supremum metric $d_{\infty}$, the Skorokhod metric $d_{0}$ and the sendograph metric $d_{S}$. Our results not only resolve certain open questions in the existing literature, but also yield completely new outcomes in terms of point-$\mathcal{A}$-transitivity.

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arXiv2026

Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree

We characterize the strictly increasing symbols $φ:\mathbb{N}_0\longrightarrow\mathbb{N}_0$ whose composition operators $C_φ$ satisfy the Frequent Hypercyclicity Criterion on the little Lipschitz space $\mathcal{L}_0(\mathbb{N}_0)$. With this result we continue the recent research about this kind of spaces and operators, but our approach relies on establishing a natural isomorphism between the Lipschitz-type spaces over rooted trees and the classical spaces $\ell^{\infty}$ and $c_0$. Such isomorphism provides an alternative framework that simplifies and allows to improve many previous results about these spaces and the operators defined there.

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arXiv2025

Questions in linear recurrence: From the $T\oplus T$-problem to lineability

We study, for a continuous linear operator $T$ on an F-space $X$, when the direct sum operator $T\oplus T$ is recurrent on $X\oplus X$. In particular: we establish, for recurrence, the analogous notion to that of (topological) weak-mixing for transitivity/hypercyclicity, namely quasi-rigidity; and we construct a recurrent but not quasi-rigid operator on each infinite-dimensional Banach space, solving the $T\oplus T$-recurrence problem in the negative way. The quasi-rigidity notion is closely related to the dense lineability of the set of recurrent vectors, and using similar conditions we study the lineability and dense lineability properties for the set of $\mathcal{F}$-recurrent vectors. This document has been split into two already published papers: Part I - Questions in linear recurrence I: The $T\oplus T$-recurrence problem. Analysis and Mathematical Physics, Volume 15, article number 1, (2025). https://doi.org/10.1007/s13324-024-00999-8 Part II - Questions in linear recurrence II: Lineability properties. Banach Journal of Mathematical Analysis, Volume 19, article number 61, (2025). https://doi.org/10.1007/s43037-025-00448-z

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arXiv2025

Shifts on trees versus classical shifts in chain recurrence

We construct continuous (and even invertible) linear operators acting on Banach (even Hilbert) spaces whose restrictions to their respective closed linear subspaces of chain recurrent vectors are not chain recurrent operators. This construction completely solves in the negative a problem posed by Nilson C. Bernardes Jr. and Alfred Peris on chain recurrence in Linear Dynamics. In particular: we show that the non-invertible case can be directly solved via relatively simple weighted backward shifts acting on certain unrooted directed trees; then we modify the non-invertible counterexample to address the invertible case, but falling outside the class of weighted shift operators; and we finally show that this behaviour cannot be achieved via classical (unilateral neither bilateral) weighted backward sifts (acting on $\mathbb{N}$ and $\mathbb{Z}$ respectively) by noticing that a classical shift is a chain recurrent operator whenever it admits a non-zero chain recurrent vector.

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arXiv2025

Recurrence in collective dynamics: From the hyperspace to fuzzy dynamical systems

We study for a dynamical system $f:X\longrightarrow X$ some of the principal topological recurrence-kind properties with respect to the induced maps $\overline{f}:\mathcal{K}(X)\longrightarrow\mathcal{K}(X)$, on the hyperspace of non-empty compact subsets of $X$, and $\hat{f}:\mathcal{F}(X)\longrightarrow\mathcal{F}(X)$, on the space of normal fuzzy sets consisting of the upper-semicontinuous functions $u:X\longrightarrow [0,1]$ with compact support and such that $u^{-1}(\{1\})\neq\varnothing$. In particular, we characterize the properties of topological and multiple recurrence for the extended systems $(\mathcal{K}(X),\overline{f})$ and $(\mathcal{F}(X),\hat{f})$, which cover the cases of the so-called nonwandering and Van der Waerden systems. Special attention is given to the case where the underlying space is completely metrizable, for which we obtain some stronger point-recurrence equivalences.

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arXiv2025

Two remarks on the set of recurrent vectors

We solve in the negative two open problems, related to the linear and topological structure of the set of recurrent vectors, asked by Sophie Grivaux, Alfred Peris and the first author of this paper. Firstly, we show that there exist recurrent operators whose set of recurrent vectors is not dense lineable; and secondly, we construct operators which are reiteratively recurrent and cyclic, but whose set of reiteratively recurrent vectors is meager.

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arXiv2024

Invariant measures from locally bounded orbits

Motivated by recent investigations of Sophie Grivaux and Étienne Matheron on the existence of invariant measures in Linear Dynamics, we introduce the concept of locally bounded orbit for a continuous linear operator $T:X\longrightarrow X$ acting on a Fréchet space $X$, and we use this new notion to construct (non-trivial) $T$-invariant probability Borel measures on $(X,\mathscr{B}(X))$.

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arXiv2024

Recurrent subspaces in Banach spaces

We study the spaceability of the set of recurrent vectors $\text{Rec}(T)$ for an operator $T:X\longrightarrow X$ on a Banach space $X$. In particular: we find sufficient conditions for a quasi-rigid operator to have a recurrent subspace; when $X$ is a complex Banach space we show that having a recurrent subspace is equivalent to the fact that the essential spectrum of the operator intersects the closed unit disc; and we extend the previous result to the real case. As a consequence we obtain that: a weakly-mixing operator on a real or complex separable Banach space has a hypercyclic subspace if and only if it has a recurrent subspace. The results exposed exhibit a symmetry between the hypercyclic and recurrence spaceability theories showing that, at least for the spaceable property, hypercyclicity and recurrence can be treated as equals.

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arXiv2022

Recurrence properties for linear dynamical systems: An approach via invariant measures

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector $x_0$, for an adjoint operator $T$ on a separable dual Banach space $X$, one can construct a $T$-invariant probability measure which contains $x_0$ in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.

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arXiv2022

Frequently recurrent operators

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family $\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-hypercyclicity.

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