Alexander Arbieto

Lp-GENERIC COCYCLES HAVE ONE-POINT LYAPUNOV SPECTRUM

We show that the sum of the first k Lyapunov exponents of linear cocycles is an upper semicontinuous function in the Lp topologies, for any 1 ≤ p ≤ ∞ and k. This fact, together with a result from Arnold and Cong, implies that the Lyapunov exponents of the Lp-generic cocycle, p < ∞, are all equal.

Equilibrium states for random non-uniformly expanding maps

We show that for a robust (C2-open) class of random non-uniformly expanding maps there exist equilibrium states for a large class of potentials. In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu–Zhao. The proof of the main result uses an extension of techniques in recent works by Alves–Araujo, Alves–Bonatti–Viana and Oliveira.

Flows with the (asymptotic) average shadowing property on three-dimensional closed manifolds

We show that the interior of the set of flows with the (asymptotic) average shadowing property on a three-dimensional closed manifold is formed by transitive Anosov flows.

Sectional Lyapunov exponents

We define sectional Lyapunov exponents and use them to characterize sectional Anosov flows in terms of dominated splittings. In particular we improve a result of Sataev.

On Weakly Hyperbolic Iterated Function Systems

We study weakly hyperbolic iterated function systems on compact metric spaces, as defined by Edalat (Inform Comput 124(2):182–197, 1996), but in the more general setting of compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure. We also define weakly hyperbolic iterated function systems for complete metric spaces and compact parameter space, extending the above mentioned definition. Furthermore, we study the question of existence of attractors in this setting. Finally, we prove a version of the results by Barnsley and Vince (Ergodic Theory Dyn Syst 31(4):1073–1079, 2011), about drawing the attractor (the so-called the chaos game), for compact parameter space.

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

Decidability of Chaos for Some Families of Dynamical Systems

C^1

On various types of shadowing for geometric Lorenz flows

generic flows whose singularities accumulated by periodic orbits have codimension one. Our result improves \cite{mp1}.

MIXING-LIKE PROPERTIES FOR SOME GENERIC AND ROBUST DYNAMICS

Abstract We show that the existence of positive Lyapounov exponents and/or SRB measures are undecidable (in the algorithmic sense) properties within some parametrized families of interesting dynamical systems: the quadratic family and Hénon maps. Because the existence of positive exponents (or SRB measures) is, in a natural way, a manifestation of “chaos,” these results may be understood as saying that the chaotic character of a dynamical system is undecidable. Our investigation is directly motivated by questions asked by Carleson and Smale in this direction.

On persistently positively expansive maps

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.