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Pablo G. Barrientos

Publications and source records attributed to Pablo G. Barrientos.

32 records · Page 2Linked to original sources
arXiv2021

Robust degenerate unfoldings of cycles and tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

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arXiv2019

Historic behavior in non-hyperbolic homoclinic classes

We show that $C^1$-generically for diffeomorphisms of manifolds of dimension $d\geq3$, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior.

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arXiv2019

Chaos near a reversible homoclinic bifocus

We show that any neighborhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on $N$-symbols for all $N\geq 2$. This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of super-homoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly rich structure.

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arXiv2018

Ergodicity of non-autonomous discrete systems with non-uniform expansion

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $C^{1+α}$ local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal $C^{1+α}$ local diffeomorphisms of a compact manifold is Lebesgue ergodic.

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arXiv2018

Expanding actions: minimality and ergodicity

We prove that every expanding minimal semigroup action of $C^1$ diffeomorphisms of a compact manifold (resp. $C^{1+α}$ conformal) is robustly minimal (resp. ergodic with respect to Lebesgue measure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.

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arXiv2018

On the chaos game of Iterated Function Systems

Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.

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arXiv2017

Robust cycles and tangencies of large codimension

We construct partially hyperbolic diffeomorphisms having semi-local robustly transitive sets with $C^1$-robust cycles of any co-index. These constructions also provide a new method to create $C^2$-robust homoclinic, equidimensional and heterodimensional tangencies of large codimension in dimension $d\geq 4$. The method to generate robust homoclinic and equidimensional tangencies also works in the symplectic setting. Thus, these are mechanisms for the robust non-hyperbolicity of symplectomorphisms in higher dimension.

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arXiv2015

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.

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arXiv2015

Dynamics of iterated function systems on the circle close to rotations

We study the dynamics of iteration function systems generated by a pair of circle diffeomorphisms close to rotations in the $C^{1+\mathrm{bv}}$-topology. We characterize the obstruction to minimality and describe the limit set. In particular, there are no invariant minimal Cantor sets, which can be seen as a Denjoy/Duminy type theorem for iterated systems on the circle.

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arXiv2015

Symbolic blender-horseshoes and applications

We study partially-hyperbolic skew-product maps over the Bernoulli shift with Hölder dependence on the base points. In the case of contracting fiber maps, symbolic blender-horseshoe is defined as an invariant set which meets any almost horizontal disk in a robust sense. These invariant sets are understood as blenders with center stable bundle of any dimension. We then give necessary conditions (covering property) on an iterated function system such that the relevant skew-product has a symbolic blender-horseshoe. We use this local plug to yield robustly non-hyperbolic transitive diffeomorphisms and robust heterodimensional cycles of co-index equal to the dimension of the central direction.

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arXiv2015

Density of fiberwise orbits in minimal iterated function systems on the circle

We study the minimality of almost every orbital branch of minimal iterated function systems (IFSs). We prove that this kind of minimality holds for forward and backward minimal IFSs generated by orientation-preserving homeomorphisms of the circle. We provide new examples of iterated functions systems where this behavior persists under perturbation of the generators.

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