arXiv subjects

Régis Varão

Publications and source records attributed to Régis Varão.

At least 19 recordsLinked to original sources

arXiv2026

Extension of hypercyclic and frequently hypercyclic subspaces

We focus on the existence of large linear structures within the sets of hypercyclic and frequently hypercyclic vectors. For operators $T$ satisfying Kitai's Criterion or the Frequent Hypercyclicity Criterion, we analyze the fundamental linear space $\{f(T)x | f \in H(\mathbb{C})\}$, studied by Herrero, Bourdon, Bès, Wengenroth, and many others. We show that the set $\{f(T)x | f \in H(\mathbb{C})\}$ can be extended within $HC(T) \cup \{0\}$ or $FHC(T) \cup \{0\}$ if $x \in HC(T)$ or $x \in FHC(T)$, respectively. The extension is such that the quotient of the new space with $\{ f(T)x \mid f \in H(\mathbb{C}) \}$ has dimension $\mathfrak{c}$ (the cardinality of the continuum). Second, we prove that generically a finite-dimensional subspace contained in $HC(T) \cup \{0\}$ can be enlarged to a subspace of dimension $\mathfrak{c}$. Third, we establish sufficient conditions for extending arbitrary linear subspaces both from $HC(T) \cup \{0\}$ and $FHC(T) \cup \{0\}$ to larger subspaces of dimension $\mathfrak{c}$.

math.FA↗

arXiv2026

Dynamics of composition operators induced by odometers

We study the linear dynamics of composition operators induced by measurable transformations on finite measure spaces, with particular emphasis on operators induced by odometers. Our first main result shows that, on a finite measure space, supercyclicity of a composition operator implies hypercyclicity. This phenomenon has no analogue in several classical settings and highlights a rigidity specific to the finite-measure context. We then focus on composition operators induced by odometers and show that many dynamical properties that are distinct for weighted backward shifts collapse in this setting. In particular, for such operators, supercyclicity, Li-Yorke chaos, hypercyclicity, weak mixing, and Devaney chaos are all equivalent. In contrast to this collapse, we show that the classical equivalence between Devaney chaos and the Frequent Hypercyclicity Criterion for weighted backward shifts fails for odometers. Specifically, we construct a mixing, chaotic, and distributionally chaotic composition operator that does not satisfy the Frequent Hypercyclicity Criterion. This combination of rigidity and separation demonstrates that the dynamical behavior of composition operators induced by odometers differs sharply from that of weighted backward shifts.

math.DS↗

arXiv2025

Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $μ$ defined on the completion of the Borel $σ$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves. Then, if $f$ preserves a continuous $\mathcal{F}$-arc length system, then we only have three possibilities for the conditional measures of $μ$ along $\mathcal F$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $λ_x$ induced by the invariant arc-length system over $\mathcal F$, or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

math.DS↗

arXiv2025

The Orbit Space Approach for Piecewise Smooth Vector Fields

In this work we develop a well-defined theory of orbit spaces for piecewise smooth vector fields (PSVFs). This approach is inspired by the techniques already used in the study of endomorphisms, namely inverse limit analysis, and has been used before for PSVFs. We then apply the construction of our theory to understanding transitivity in PSVFs. Our results prove that the known examples of transitive PSVFs in the literature, the bean model and the sphere model, are indeed transitive in the orbit space.

math.DS↗

arXiv2025

Folding and Metric Entropies for Extended Shifts

In this paper we calculate the metric and folding entropies for a family of non-invertible symbolic dynamical systems $(Σ_{m_-,m_+}, σ_ϕ)$ which generalizes the standard bilateral Bernoulli shifts. The space $Σ_{m_-,m_+}$ consists of symbolic sequences over two distinct finite alphabets, with dynamics governed by a shift map $σ_ϕ$ incorporating a non-invertible function $ϕ$ that maps one of the alphabets to the other one. These systems are, for instance, particularly useful for encoding the many-to-one baker's transformation endomorphisms, and they can also be seen as a skew product with a unilateral Bernoulli shift on the base.

math.DS↗

arXiv2025

Anosov actions: minimality of foliations or suspension action

We prove that an Anosov action of $\mathbb{R}^k$ over a compact manifold $M$ transitive on regular sub-cones satisfies the dichotomy: each stable and unstable leaf is dense or the Anosov action is topologically conjugated to a suspension of a $\mathbb{Z}^k$-Anosov action. This represents an important progress toward addressing Verjovsky's extended conjecture for Anosov actions, as developed by Barbot and Maquera.

math.DS↗

arXiv2025

Entropy for compact operators and results on entropy and specification

We investigate the topological entropy of operators. More precisely, in the Banach space setting, we show that compact operators have finite entropy, which depends solely on their point spectrum. Moreover, for operators on $F$-spaces, we explore the relationship between the specification property and entropy. In particular, we show that the specification property implies infinite topological entropy, while the operator specification property implies positive entropy. We also show that the invariance principle fails for the class of compact operators.

math.DS↗

arXiv2024

Characterization of Non-Deterministic Chaos in Two-Dimensional Non-Smooth Vector Fields

Our context is Filippov systems defined on two-dimensional manifolds having a finite number of tangency points. We prove that topological transitivity is a necessary and sufficient condition for the occurrence of non-deterministic chaos when the Filippov system has non-empty sliding or escaping regions. A fundamental result for continuous flows is the equivalence of topological transitivity and existence of a dense orbit. We prove in our setting that topological transitivity for Filippov systems is indeed equivalent to the existence of a dense Filippov orbit, although, in contrast to the continuous case, we are not able to garantee that the dense orbit implies the existence of a residual set of dense orbits. Finally we prove that, in this context, topological transitivity implies strictly positive topological entropy for the Filippov system. This calculation is made using techniques similar to those from symbolic dynamics.

math.DS↗

arXiv2023

Anosov Endomorphisms on the 2-torus: Regularity of foliations and rigidity

We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the 2-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterization of smooth conjugacy between special Anosov endomorphisms and their linearizations.

math.DS↗

arXiv2021

Chain Recurrence and Positive Shadowing in Linear Dynamics

We study shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is actually a closed invariant subspace. We prove that every chain transitive linear dynamical system with the shadowing property is frequently hypercyclic, and as a corollary obtain that every shadowing hypercyclic linear dynamical system is frequently hypercyclic.

math.DS↗

arXiv2021

A note on invariant measures for Filippov systems

We are interested in Filippov systems which preserve a probability measure on a compact manifold. We define a measure to be invariant for a Filippov system as the natural analogous definition of invariant measure for flows. Our main result concerns Filippov systems which preserve a probability measure equivalent to the volume measure. As a consequence, the volume preserving Filippov systems are the refractive piecewise volume preserving ones. We conjecture that if a Filippov system admits an invariant probability measure, this measure does not see the trajectories where there is a break of uniqueness. We prove this conjecture for Lipschitz differential inclusions. Then, in light of our previous results, we analyze the existence of invariant measures for many examples of Filippov systems defined on compact manifolds.

math.DS↗

arXiv2021

There exist transitive piecewise smooth vector fields on $\mathbb{S}^2$ but not robustly transitive

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $§^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $§^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $§^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $§^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.

math.DS↗

arXiv2020

Equilibrium states for maps isotopic to Anosov

In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on $\mathbb{T}^4$, with 2-dimensional center foliation. To do so we propose to study the disintegration of measures along 1-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.

math.DS↗

arXiv2020

The disintegration of measures with periodic repetitive pattern

In the context of locally constant skew-products over the shift with circle fiber maps we introduce the notion of measures with periodic repetitive pattern, inspired by \cite{GorIlyKleNal:05} and which includes the non-hyperbolic measures they construct. We prove that these measures have atomic disintegration along the central fibers.

math.DS↗

arXiv2017

Measure Rigidity and Disintegration: Time-one map of flows

An invariant measure for a flow is, of course, an invariant measure for any of its time-t maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the time-one map to be invariant for the flow itself? We give an answer in terms of measure disintegration. Surprisingly all it takes is the measure not to be "too much pathological in the orbits". We prove the following rigidity result. If $μ$ is an ergodic probability for the time-one map of a flow, then it is either highly pathological in the orbits, or it is highly regular (i.e invariant for the flow). In particular this measure rigidity result is also true for measurable flows by the classical Ambrose-Kakutani's representation theorem for measurable flows.

math.DS↗

arXiv2016

Chaos induced by sliding phenomena in Filippov systems

In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit. More specifically we prove that the first return map, defined nearby this orbit, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.

math.DS↗

arXiv2016

Rigidity for partially hyperbolic diffeomorphisms

In this work we completely classify $C^\infty$ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^\infty$ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and $C^\infty$ center foliation.

math.DS↗

arXiv2016

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization. Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.

math.DS↗