arXiv subjects

Manuel Saavedra

Publications and source records attributed to Manuel Saavedra.

6 recordsLinked to original sources

arXiv2026

Observation Schemes and Irregularity in Linear Dynamics

We develop a structural framework for irregularity in linear dynamics centered on the space $\Upsilon$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li--Yorke chaos, mean Li--Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Ces\`aro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(\mu_m)$-irregular and distributionally $(\mu_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $\Upsilon$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.

math.FA↗

arXiv2026

On Generalized Hyperbolicity, Stability, and Shadowing for Linear Operators

This work studies the relations between shadowing, topological stability, Lipschitz structural stability, and pseudo-hyperbolicity for linear dynamical systems on Banach spaces. The main result establishes that pseudo-hyperbolicity implies both topological stability and strong Lipschitz structural stability, whereas strong Lipschitz structural stability implies the shadowing property. In addition, a spectral characterization of pseudo-hyperbolicity is obtained, yielding an equivalence between pseudo-hyperbolicity, topological stability, strong Lipschitz structural stability, and the shadowing property for invertible operators whose eigenspaces associated with unimodular eigenvalues admit closed complements. In particular, these four notions are equivalent on Hilbert spaces. Combined with a recent result of Dragi\v{c}evi\'c and Pituk, these results yield, for a large class of Banach spaces, an invertible operator that has the shadowing property but is not generalized hyperbolic, thereby answering an open problem in linear dynamics.

math.DS↗

arXiv2025

On Spaceability within Linear Dynamics

We investigate spaceability phenomena in linear dynamics from a structural perspective. Given a continuous linear operator $T:X \to X$, we introduce the set $\Omega(T)$, consisting of all continuous linear operators $h:X \to X$ for which there exists a strictly increasing sequence $(\theta_n)_n$ of positive integers such that the set $\{x \in X : \displaystyle{\lim_{n \rightarrow \infty} T^{\theta_n}x = h(x)}\}$ is dense in $X$. Within this framework, two classical phenomena--the existence of hypercyclic and recurrent subspaces in separable infinite-dimensional complex Banach spaces--emerge as instances of a common underlying structure described by $\Omega(T)$. To analyze $\Omega(T)$, we introduce the notion of collections simultaneously approximated (c.s.a.) by $T$, and show that every maximal c.s.a. is an SOT-closed affine manifold. For quasi-rigid operators on separable Banach spaces, we establish the existence of a unique maximal c.s.a. containing the identity operator. Furthermore, we examine $\Omega(T)$ through the left-multiplication operator $L_T$ acting on the algebra of bounded operators. Our approach combines two key ingredients: a refinement of A. L\'opez's technique on recurrent subspaces for quasi-rigid operators, and a common dense-lineability result obtained by the first author and A. Arbieto. These tools yield new spaceability results for the sets $\Omega(T)$, $\mathcal{AP}\Omega(T)$, and for any countable c.s.a. by $T$.

math.FA↗

arXiv2025

Super-Shadowing and Supercyclicity

We introduce the super-shadowing property in linear dynamics, where pseudotrajectories are approximated by sequences of the form $(\lambda_nT^nx)$, with $(\lambda_n)_n$ being complex scalars. For compact operators on Banach spaces, we characterize the operators that possess the positive super-shadowing property and the positive limit super-shadowing property. Additionally, we demonstrate that no surjective isometric operator on a separable Banach space $X$ with $\text{dim}(X)>1$ can exhibit the positive super-shadowing property. Finally, we provide some results on upper frequently supercyclic and reiteratively supercyclic operators.

math.FA↗

arXiv2025

Dense Lineable Criterion for Linear Dynamics

We study Li-Yorke chaos for sequences of continuous linear operators from an $F$-space to a normed space. We introduce the \emph{D-phenomenon} to establish a common dense lineable criterion that encompasses properties such as recurrence, universality, and Li-Yorke chaos. We show that in every infinite-dimensional separable complex Banach space, there exists a sequence of operators with a dense set of irregular vectors but without a dense irregular manifold, and we exhibit a recurrent operator whose set of recurrent vectors is not dense-lineable. This resolves in the negative a question posed by Grivaux et al.

math.FA↗

arXiv2024

Quasi-rigid operators and hyper-recurrence

We study recurrent operators from a new perspective by introducing the notion of hyper-recurrent operators and establish robust connections with quasi-rigid operators. For example, we prove that a recurrent operator on a separable Banach space is quasi-rigid if and only if it is a linear factor of a hyper-recurrent operator, and show that the quasi-rigid operators found in Costakis, Manoussos and Parissis's work, along with many others, are, in fact, hyper-recurrent operators. Furthermore, we provide a negative answer, using a class of operators introduced by Tapia, to the question by Costakis et al. whether $T \oplus T$ is recurrent whenever $T$ is.

math.FA↗