Searcharxiv⌕ Search

arXiv subjects

A. Arbieto

Publications and source records attributed to A. Arbieto.

13 recordsLinked to original sources
arXiv2026

On the Expansiveness of Invariant Measures under Pseudogroups

In this paper, we define and study weak expansive and expansive measures for pseudogroups, these two notions appear when analyzing the role of the generating set. We investigate the relations between such properties. We also provide a criterion for a measure to be weak expansive through the positivity of its entropy, generalizing the work of Arbieto and Morales. We also show that in some settings equicontinuous pseudogroups have no expansive measures.

math.DS↗
arXiv2026

Robust transitivity without sectional-hyperbolicity

For any integer $n \geq 5$, we construct an $n$-dimensional $C^1$ vector field exhibiting a robustly transitive singular attractor which is not sectional-hyperbolic. Nevertheless, the attractor is singular-hyperbolic. This provides the first such examples improving some features of the constructions in [17, 32].

math.DS↗
arXiv2020

Shadowing for codimension one sectional-Anosov flows

In hyperbolic dynamics, a well-known result is that every hyperbolic attracting set, have a finite pseudo-orbit tracing property (FPOTP). It's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics; Komuro in [Lorenz attractors do not have the pseudo-orbit tracing property], provides a negative answer for this question, by proving that the geometric Lorenz Attractor doesn't have a FPOTP. In this paper, we generalized the result of Komuro, we prove that every codimension one sectional-hyperbolic attractor set with a unique singularity Lorenz-like, which is of boundary-type, does not have FPOTP.

math.DS↗
arXiv2019

Positive Entropy Through Pointwise Dynamics

We define some pointwise properties of topological dynamical systems and give pointwise conditions for such a system possesses positive topological entropy. We give sufficient conditions to obtain positive topological entropy for maps which are approximated by maps with the shadowing property in a uniform way.

math.DS↗
arXiv2013

Existence of attractors, homoclinic tangencies and singular hyperbolicity for flows

We prove that every $C^1$ generic three-dimensional flow has either infinitely many sinks, or, infinitely many hyperbolic or singular-hyperbolic attractors whose basins form a full Lebesgue measure set. We also prove in the orientable case that the set of accumulation points of the sinks of a $C^1$ generic three-dimensional flow has no dominated splitting with respect to the linear Poincaré flow. As a corollary we obtain that every three-dimensional flow can be $C^1$ approximated by flows with homoclinic tangencies or by singular-Axiom A flows.

math.DS↗
arXiv2013

On Various Types of Shadowing for Geometric Lorenz Flows

We show that Lorenz flows have neither limit shadowing property nor average shadowing property nor the asymptotic average shadowing property where the reparametrizations related to these concepts relies on the set of increasing homeomorphisms with bounded variation.

math.DS↗
arXiv2013

Some properties of surface diffeomorphisms

We obtain some properties of $C^1$ generic surface diffeomorphisms as finiteness of {\em non-trivial} attractors, approximation by diffeomorphisms with only a finite number of {\em hyperbolic} homoclinic classes, equivalence between essential hyperbolicity and the hyperbolicity of all {\em dissipative} homoclinic classes (and the finiteness of spiral sinks). In particular, we obtain the equivalence between finiteness of sinks and finiteness of spiral sinks, abscence of domination in the set of accumulation points of the sinks, and the equivalence between Axiom A and the hyperbolicity of all homoclinic classes. These results improve \cite{A}, \cite{a}, \cite{m} and settle a conjecture by Abdenur, Bonatti, Crovisier and Díaz \cite{abcd}.

math.DS↗
arXiv2012

Symbolic Extensions and dominated splittings for Generic C^1-Diffeomorphisms

Let Diff^1(M) be the set of all C^1-diffeomorphisms f : M \rightarrow M, where M is a compact boundaryless d-dimensional manifold, d \geq 2. We prove that there is a residual subset R of Diff^1(M) such that if f \in R and if H(p) is the homoclinic class associated with a hyperbolic periodic point p, then either H(p) admits a dominated splitting of the form E\oplusF1 \oplus...\oplusFk \oplusG, where Fi is not hyperbolic and one-dimensional, or f|H(p) has no symbolic extensions.

math.DS↗
arXiv2012

Lyapunov stability and sectional-hyperbolicity for higher-dimensional flows

We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with residual basin of attraction. This represents a partial positive answer to conjectures in \cite{am}, the Palis conjecture \cite{pa} and extend the Araujo's thesis to higher dimensions \cite{a}.

math.DS↗
arXiv2011

Expansivity of ergodic measures with positive entropy

We prove that for every ergodic invariant measure with positive entropy of a continuous map on a compact metric space there is $δ>0$ such that the dynamical $δ$-balls have measure zero. We use this property to prove, for instance, that the stable classes have measure zero with respect to any ergodic invariant measure with positive entropy. Moreover, continuous maps which either have countably many stable classes or are Lyapunov stable on their recurrent sets have zero topological entropy. We also apply our results to the Li-Yorke chaos.

math.DS↗
arXiv2011

A dichotomy for higher dimensional flows

We analyze the dichotomy between {\em sectional-Axiom A flows} (c.f. \cite{memo}) and flows with points accumulated by periodic orbits of different indices. Indeed, this is proved for $C^1$ generic flows whose singularities accumulated by periodic orbits have codimension one. Our result improves \cite{mp1}.

math.DS↗
arXiv2009

Mixing rate for semi-dispersing billiards with non-compact cusps

Since the seminal work of Sinai one studies chaotic properties of planar billiards tables. Among them is the study of decay of correlations for these tables. There are examples in the literature of tables with exponential and even polynomial decay. However, until now nothing is known about mixing properties for billiard tables with non-compact cusps. There is no consensual definition of mixing for systems with infinite invariant measure. In this paper we study geometric and ergodic properties of billiard tables with a non-compact cusp. The goal of this text is, using the definition of mixing proposed by Krengel and Sucheston for systems with invariant infinite measure, to show that the billiard whose table is constituted by the x-axis and and the portion in the plane below the graph of $f(x)=\frac{1}{x+1}$ is mixing and the speed of mixing is polynomial.

math.DS↗