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Enrique Pujals

Publications and source records attributed to Enrique Pujals.

26 records · Page 2Linked to original sources
arXiv2013

Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale's Axiom A property is $C^1$-dense among the systems in this family, despite the existence of $C^2$-open subsets (closely related to the so-called Newhouse phenomena) where Smale's Axiom A is violated. In particular, this provides some evidence towards Smale's conjecture that Axiom A is a $C^1$-dense property among surface diffeomorphisms. The basic tools in the proof of this result are: 1) a recent theorem of Moreira saying that stable intersections of dynamical Cantor sets (one of the main obstructions to Axiom A property for surface diffeomorphisms) can be destroyed by $C^1$-perturbations; 2) the good geometry of the dynamical critical set (in the sense of Rodriguez-Hertz and Pujals) thanks to the particular form of Benedicks-Carleson toy models.

math.DS↗
arXiv2013

Curvature and partial hyperbolicity

Using quadratic forms, we stablish a criteria to relate the curvature of a Riemannian manifold and partial hyperbolicity of its geodesic flow. We show some examples which satisfy the criteria and another which does not satisfy it but still has a partially hyperbolic geodesic flow.

math.DS↗
arXiv2013

Ergodic properties of skew products in infinite measure

Let (Ω,μ) be a shift of finite type with a Markov probability, and (Y,ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a measure-preserving automorphism of (Y,ν). We study skew products of the form (ω,y) --> (σω,Φ_{ω_0}(y)), where σ=shift map on (Ω,μ). We prove that, when the skew product is conservative, it is ergodic if and only if the Φ_i's have no common non-trivial invariant set. In the second part we study the skew product when Ω={0,1}^Z, μ=Bernoulli measure, and Φ_0,Φ_1 are R-extensions of a same uniquely ergodic probability-preserving automorphism. We prove that, for a large class of roof functions, the skew product is rationally ergodic with return sequence asymptotic to \sqrt{n}, and its trajectories satisfy the central, functional central and local limit theorem.

math.DS↗
arXiv2012

Law of large numbers for certain cylinder flows

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic and rationally ergodic along a subsequence of iterates. In particular, they exhibit law of large numbers. This is accomplished by explicitly calculating, for a subsequence of iterates, the number of visits to zero, and it is shown that such number has a gaussian distribution.

math.DS↗
arXiv2012

Robust Transitivity for Endomorphisms

We address the problem about under what conditions an endomorphism having a dense orbit, verifies that a sufficiently close perturbed map also exhibits a dense orbit. In this direction, we give sufficient conditions, that cover a large class of examples, for endomorphisms on the n-dimensional torus to be robustly transitive: the endomorphism must be volume expanding and any large connected arc must contain a point such that its future orbit belong to an expanding region.

math.DS↗
arXiv2007

Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $λ$-Aluthge transform

Let $λ\in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ Δ_λ(T )= |T|^λ U |T |^{1-λ}. $$ Let $Δ_λ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ_λ^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map $(\la, T) \mapsto \alulit{\infty}{T}$, and we study for which matrices the map $(0,1)\ni λ\mapsto Δ_λ^{\infty}(T)$ is constant.

math.FA↗
arXiv2007

Singular-hyperbolic attractors are chaotic

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their orbits coincide. Secondly, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a $u$-Gibbs state and an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows.

math.DS↗