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Alexander Arbieto

Publications and source records attributed to Alexander Arbieto.

At least 19 recordsLinked to original sources
arXiv2026

On the Abundance of Phase Transitions in Dynamical Systems

In this work, we investigate the mechanisms that trigger phase transitions in both discrete-time and continuous-time dynamical systems. We prove a simple topological condition that implies the existence of H\"older continuous potentials displaying phase transitions. As a consequence, we show that phase transitions are typical in several scenarios. Specifically, for $C^1$ diffeomorphisms and $C^1$ vector fields, we show that phase transitions are generic among non-transitive systems. In low dimensions, we conclude that they are generic for non-Anosov systems. Finally, in the $C^0$ setting, we prove that on any compact topological manifold of dimension different from 4, there exists a dense set of homeomorphisms with finite entropy that display phase transitions.

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arXiv2025

Entropy Flexibility of Dynamical Systems

Inspired by Katok's intermediate entropy property [Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137-173], we introduce and study the notion of entropy flexibility for discrete-time and continuous-time dynamical systems. By using renewal systems techniques, we show that this property is present in several classes of systems where any intermediate value of entropy can be attained on a strictly ergodic sub-system. In addition, we prove an entropy flexibility analogue of Katok's conjecture: Entropy flexibility is a typical property for vector fields on 3-manifolds and surface diffeomorphisms.

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arXiv2025

Existence and Finiteness of equilibrium states for some Partially hyperbolic endomorphisms

We establish the existence and finiteness of equilibrium states for a class of partially hyperbolic endomorphisms. In our first result, we assume that the central direction is simple. In the second result, we consider the case where there exists a dominated splitting along the central direction, which is decomposed into one-dimensional subbundles. This latter result extends the work of C. Alvarez and M. Cantarino \cite{alvarez2022existence} to higher-dimensional central directions. Finally, we demonstrate the finiteness of measures of maximal entropy under the assumption that the central direction is one-dimensional and the integrated Lyapunov exponent is bounded away from zero.

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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.

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arXiv2024

Contributions to the Theory of Asymptotically Sectional Hyperbolic Flows

In this paper, we make several contributions to the theory of asymptotically sectional-hyperbolic (ASH) flows. First, we prove that every star ASH attractor for a $C^2$ vector field is, in fact, sectional-hyperbolic (SH). Second, we establish that all ASH attractors exhibit the property of entropy flexibility. Additionally, we show that any ASH attractor for three-dimensional vector fields is entropy-expansive and admits periodic orbits. Finally, we provide a lower bound for the growth rate of periodic orbits in an ASH attractor.

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arXiv2024

Rescaled-Expansive Flows: Unstable Sets and Topological Entropy

In this work we introduce and explore a rescaled-theory of local stable and unstable sets for rescaled-expansive flows and its applications to topological entropy. We introduce a rescaled version of the local unstable sets and the unstable points. We find conditions for points of the phase space to exhibit non-trivial connected pieces of such unstable sets. We apply these results to the problem of proving positive topological entropy for rescaled-expansive flows with non-singular Lyapunov stable sets.

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arXiv2023

On the Shadowableness of Flows With Hyperbolic Singularities

In this work we study the existence of singular flows satisfying shadowing-like properties. More precisely, we prove that if C1 -vector field on a closed manifold induces a chain-recurrent flow containing an attached hyperbolic singularity of stable or unstable index-one, then this flow cannot satisfy the shadowing property. If the manifold is non-compact, the vector field is complete and non-wandering, we prove that we prove that the existence of index-one hyperbolic singularities prevents the induced flow to satisfy the rescaled-shadowing property introduced in [6].

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arXiv2022

Expansive Lie Group Actions

In this work we introduce a concept of expansiveness for actions of connected Lie groups. We study some of its properties and investigate some implications of expansiveness. We study the centralizer of expansive actions and introduce CW-expansiveness for pseudo-group actions. As an application, we prove positiveness of geometric entropy for expansive foliations and expansive group actions.

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arXiv2022

Shadowing, topological entropy and recurrence of induced Morse-Smale diffeomorphisms

Let $f : M \rightarrow M$ be a Morse-Smale diffeomorphism defined on a compact and connected manifold without boundary. Let $C(M)$ denote the hyperspace of all subcontinua of M endowed with the Hausdorff metric and $C(f) : C(M) \rightarrow C(M)$ denote the induced homeomorphism of $f$. We show in this paper that if $M$ is the unit circle $S^1$ then the induced map $C(f)$ has not the shadowing property. Also we show that the topological entropy of $C(f)$ has only two possible values: $0$ or $\infty$. In particular, we show that the entropy of $C(f)$ is $0$ when $M$ is the unit circle $S^1$ and it is $\infty$ if the dimension of the manifold $M$ is greater than two. Furthermore, we study the recurrence of the induced maps $2^f$ and $C(f)$ and sufficient conditions to obtain infinite topological entropy in the hyperspace.

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arXiv2020

On The Entropy of Continuous Flows With Uniformly Expansive Points and The Globalness of Shadowable Points With Gaps

In this work we study the problem of positiveness of topological entropy for flows using pointwise dynamics. We show that the existence of a non-periodic nonwandering point of an expansive and non-singular flow with shadowing is a sufficient condition to to obtain positive topological entropy. Moreover, we can deal with flows with singularities, showing that the existence of a non-wandering, non-critical, strongly-shadowable, and uniform-expansive point implies the existence of a symbolic subshift. Finally, we discuss pointwise versions of some shadowing-type properties.

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arXiv2020

On the Space of Iterated Function Systems and Their Topological Stability

We study iterated function systems (IFS) with compact parameter space. We show that the space of IFS with phase space $X$ is the hyperspace of the space of self continuous maps of $X$. With this result we obtain that the Hausdorff distance is a natural metric for this space which we use to define topological stability. Then we prove, in the context of IFS, the classical results showing that shadowing property is a necessary condition for topological stability and shadowing property added to expansiveness are a sufficient condition for topological stability. To prove these statements, in fact, we use a stronger type of shadowing, called concordant shadowing property. We also give an example showing that concordant shadowing property is truly different than the traditional definition of shadowing property for IFS.

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arXiv2020

Transfer operators and atomic decomposition

We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones.

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arXiv2016

On weakly hyperbolic iterated functions systems

We study weakly hyperbolic iterated function systems on compact spaces, as defined by Edalat, but in the more general setting of a compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint, and prove that the measure theoretical attractor is ergodic. We also define weakly hyperbolic iterated functions systems for complete spaces and compact parameter space, and prove that this definition extends the one given by Edalat. Furthermore, we study the question of existence of the attractors in this setting. Finally, we prove a version of the results of Barnsley and Vince about drawing the attractor (also called the chaos game), for the case of compact parameter space.

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arXiv2016

Generic properties of magnetic flows

We obtain Kupka-Smale theorem and Franks lemma for magnetic flows on manifolds with any dimension. This improves Miranda result on surfaces. However our methods relies on geometric control theory, like in Rifford and Ruggiero articles.

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arXiv2015

On m-minimal partially hyperbolic diffeomorphisms

We discuss about the denseness of the strong stable and unstable manifolds of partially hyperbolic diffeomorphisms. In this sense, we introduce a concept of m-minimality. More precisely, we say that a partially hyperbolic diffeomorphisms is m-minimal if m-almost every point in M has its strong stable and unstable manifolds dense in M. We show that this property has dynamics consequences: topological and ergodic. Also, we prove the abundance of m-minimal partially hyperbolic diffeomorphisms in the volume preserving and symplectic scenario.

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arXiv2015

CW-expansivity and entropy for flows

We define the concept of continuum wise expansive for flows, and we prove that continuum wise expansive flows on compact metric spaces with topological dimension greater than one have positive entropy.

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arXiv2014

Mixing-like properties for some generic and robust dynamics

We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robustly topologically mixing.

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