Opening Gaps in the Spectrum of Strictly Ergodic Schr\"odinger Operators (2009)
Avila, Artur, Bochi, Jairo, Damanik, David
We consider Schr\"odinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the...
Avila, Artur, Bochi, Jairo, Wilkinson, Amie
We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns--Wilkinson and Avila--Santamaria--Viana. Combining this new...
Some Characterizations of Domination (2008)
Bochi, Jairo, Gourmelon, Nicolas
We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\Sigma$ in $GL(d,\mathbb{R})$ with...
Uniformly Hyperbolic Finite-Valued SL(2,R)-Cocycles (2008)
Avila, Artur, Bochi, Jairo, Yoccoz, Jean-Christophe
We consider finite families of SL(2,R) matrices whose products display uniform exponential growth. These form open subsets of (SL(2,R))^N, and we study their components, boundary, and complement. We...
We prove that if $f$ is a $C^1$-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov...
Pisa Lectures on Lyapunov Exponents (2007)
These notes cover the course we taught at the research trimester on Dynamical Systems organized by Stefano Marmi at the Centro di Ricerca Matematica Ennio di Giorgi/Scuola Normale Superiore di Pisa...
Avila, Artur, Bochi, Jairo, Damanik, David
We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle...
A uniform dichotomy for generic $SL(2,R)$ cocycles over a minimal base (2006)
We consider continuous $SL(2,R)$-cocycles over a minimal homeomorphism of a compact set $K$ of finite dimension. We show that the generic cocycle either is uniformly hyperbolic or has uniform...
Generic expanding maps without absolutely continuous invariant $\sigma$-finite measure (2006)
We show that a $C^1$-generic expanding map of the circle has no absolutely continuous invariant $\sigma$-finite measure.
A generic $C^1$ map has no absolutely continuous invariant probability measure (2006)
Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with...
Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles (2005)
We consider the linear cocycle $(T,A)$ induced by a measure preserving dynamical system $T:X \to X$ and a map $A:X \to \mathit{SL}(2,\mathbb{R})$. We address the dependence of the upper Lyapunov...
The Lyapunov exponents of generic volume-preserving and symplectic maps (2005)
We show that the integrated Lyapunov exponents of $C^1$ volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is...
The Lyapunov exponents of generic volume preserving and symplectic systems (2005)
We show that the integrated Lyapunov exponents of C1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial...
A remark on conservative diffeomorphisms (2004)
Bochi, Jairo, Fayad, Bassam, Pujals, Enrique
We show that a stably ergodic diffeomorphism can be $C^1$ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.
$L^p$-generic cocycles have one-point Lyapunov spectrum (2002)
Arbieto, Alexander, Bochi, Jairo
We show the sum of the first $k$ Lyapunov exponents of linear cocycles is an upper semicontinuous function in the $L^p$ topologies, for any $1 \le p \le \infty$ and $k$. This fact, together with a...
The Lyapunov exponents of generic volume preserving and symplectic systems (2002)
We show that the integrated Lyapunov exponents of $C^1$ volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is...
Inequalities for numerical invariants of sets of matrices (2002)
We prove three inequalities relating some invariants of sets of matrices, such as the joint spectral radius. One of the inequalities, in which proof we use geometric invariant theory, has the...
Genericity of zero Lyapunov exponents (2002)
We show that, for any compact surface, there is a residual (dense $G_\delta$) set of $C^1$ area preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponents a.e. This result was...
Robust transitivity and topological mixing for $C^1$-flows (2002)
Abdenur, Flavio, Avila, Artur, Bochi, Jairo
We prove that non-trivial homoclinic classes of $C^r$-generic flows are topologically mixing. This implies that given $\Lambda$ a non-trivial $C^1$-robustly transitive set of a vector field $X$,...
A formula with some applications to the theory of Lyapunov exponents (2001)
We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub,...