Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension
Artur Avila
We show that given a fixed irrational rotation of the $d$-dimensional torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [6] and Fayad-Krikorian [5]. The key technique is the analyticity of $m$-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten-Martini Problem [2].
keywords: quasiperiodic cocycles. Lyapunov exponents
Uniform exponential growth for some SL(2, R) matrix products
Artur Avila Thomas Roblin
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs less than $o(\frac{n}{\log n\log\log n})$ times.
keywords: hyperbolicity R) Lyapounov exponent. matrix products SL(2
Smoothness of solenoidal attractors
Artur Avila Sébastien Gouëzel Masato Tsujii
We consider dynamical systems generated by skew products of affine contractions on the real line over angle-multiplying maps on the circle $S^1$:

$\ T:S^{1}\times \R\to S^{1}\times \R,\qquad T(x,y)=(l x, \lambda y+f(x)) \

where l ≥ 2, $0<\lambda<1$ and $f$ is a $C^{r}$ function on $S^{1}$. We show that, if $\lambda^{1+2s}l>1$ for some $0\leq s< r-2$, the density of the SBR measure for $T$ is contained in the Sobolev space $W^{s}(S^{1}\times \R)$ for almost all ($C^r$generic, at least) $f$.

keywords: invariant measure. Hyperbolic attractor Anosov endomorphism absolutely continuous

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