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Enrique R. Pujals

Publications and source records attributed to Enrique R. Pujals.

15 recordsLinked to original sources
arXiv2020

The Evolutionary Robustness of Forgiveness and Cooperation

We study the evolutionary robustness of strategies in infinitely repeated prisoners' dilemma games in which players make mistakes with a small probability and are patient. The evolutionary process we consider is given by the replicator dynamics. We show that there are strategies with a uniformly large basin of attraction independently of the size of the population. Moreover, we show that those strategies forgive defections and, assuming that they are symmetric, they cooperate.

math.DS↗
arXiv2020

Orders of Growth and Generalized Entropy

We construct the complete set of orders of growth and we define on it the generalized entropy of a dynamical systems. With this object we provide a framework where we can study the separation of orbits of a map beyond the scope of exponential growth. We are going to show that this construction is particularly useful to study families of dynamical systems with vanishing entropy. Moreover, we are going to see that the space of orders of growth in which orbits are separated is wilder than expected. This is going to be achieved with different types of examples.

math.DS↗
arXiv2016

Expansivity and Shadowing in Linear Dynamics

In the early 1970's Eisenberg and Hedlund investigated relationships between expansivity and spectrum of operators on Banach spaces. In this paper we establish relationships between notions of expansivity and hypercyclicity, supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space is $c_0$ or $\ell_p$ ($1 \leq p < \infty$), we give complete characterizations of weighted shifts which satisfy various notions of expansivity. We also establish new relationships between notions of expansivity and spectrum. Moreover, we study various notions of shadowing for operators on Banach spaces. In particular, we solve a basic problem in linear dynamics by proving the existence of nonhyperbolic invertible operators with the shadowing property. This also contrasts with the expected results for nonlinear dynamics on compact manifolds, illuminating the richness of dynamics of infinite dimensional linear operators.

math.DS↗
arXiv2014

Hölder stability for $C^r$ central translations

We consider the class of diffeomorphisms of a manifold that its differential keeps invariant a one-dimensional subbundle $E$. For that type of diffeomorphisms is naturally defined a one-parameter family called $E-$translation. We prove that if a diffeomorphisms in above mentioned class is conjugate to its $E-$translation and the conjugacy is at distance $α$-Hölder to the identity respect to the parameter and $α>1/2$, then the $E$-direction is hyperbolic. This theorem is also sharp as it is be discussed with some examples. We also deal with the continuously stable case in the Skew-Products context with one-dimensional fibers, requiring extra hypothesis along the fibers like either non-negative second derivative or negative Schwartzian.

math.DS↗
arXiv2013

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.

math.DS↗
arXiv2012

Some Consequences of the Shadowing Property in Low Dimensions

We consider low-dimensional systems with the shadowing property. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $ε$-transitive class, and in contrast we provide an example of a $C^\infty$ Kupka-Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally we consider the case of transitive endomorphisms of the circle, and we prove that the $α$-Hölder shadowing property with $α>1/2$ implies that the system is conjugate to an expanding map.

math.DS↗
arXiv2011

Robust Transitivity in Hamiltonian Dynamics

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce $C^r$ open sets ($r=1, 2, ..., \infty$) of symplectic diffeomorphisms and Hamiltonian systems, exhibiting "large" robustly transitive sets. We show that the $C^\infty$ closure of such open sets contains a variety of systems, including so-called a priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of the proof is a combination of studying minimal dynamics of symplectic iterated function systems and a new tool in Hamiltonian dynamics which we call symplectic blender.

math.DS↗
arXiv2009

C^k-Robust transitivity for surfaces with boundary

We prove that C^1-robustly transitive diffeomorphisms on surfaces with boundary do not exist, and we exhibit a class of diffeomorphisms of surfaces with boundary which are C^k-robustly transitive, with k greater or equal than 2. This class of diffeomorphisms are examples where a version of Palis' conjecture on surfaces with boundary, about homoclinic tangencies and uniform hyperbolicity, does not hold in the C^2-topology. This follows showing that blow-up of pseudo-Anosov diffeomorphisms on surfaces without boundary, become C^2-robustly topologically mixing diffeomorphisms on a surfaces with boundary.

math.DS↗
arXiv2007

The iterated Aluthge transforms of a matrix converge

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ Δ(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $Δ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $Δ^{0}(T)=T$ and $Δ^{n}(T)=Δ(Δ^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ matrix $T$. This result was conjecturated by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.

math.FA↗
arXiv2005

Motion of vortices implies chaos in Bohmian mechanics

Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.

quant-ph↗
arXiv2003

Dynamical zeta functions for analytic surface diffeomorphisms with dominated splitting

We study the Ruelle dynamical determinant of a real analytic diffeomorphism on a compact surface, assuming that the tangent space over the nonwandering set admits a dominated splitting. Combining previous work of Pujals and Sambarino with methods introduced by Rugh, we show that the determinant is either entire or holomorphic in a (possibly multiply) slit plane.

math.DS↗
arXiv2003

Tangent bundles dynamics and its consequences

We will consider here some dynamics of the tangent map, weaker than hyperbolicity, and we will discuss if these structures are rich enough to provide a good description of the dynamics from a topological and geometrical point of view. This results are useful in attempting to obtain global scenario in terms of generic phenomena relative both to the space of dynamics and to the space of trajectories. Moreover, we will relate these results with the study of systems that remain globally transitive under small perturbations.

math.DS↗