Bassam Fayad

Constructions in elliptic dynamics

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago ( Trans. Moscow Math. Soc. 23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi ( Asterisque 49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere ( J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author ( Ergod. Th. & Dynam. Sys. 22 (2002) 437–468; Proc. Amer. Math. Soc. 130 (2002), 103–109).

A KAM SCHEME FOR SL(2, R) COCYCLES WITH LIOUVILLEAN FREQUENCIES

We develop a new KAM scheme that applies to SL(2,

Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2, ℝ) cocycles

{{\mathbb R}}

Rank one and mixing differentiable flows

) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg–Sinai’s theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

Exponential growth of product of matrices in {\rm SL}(2,{\mathbb{R}})

In this paper, we study the display of weak mixing by reparameterized linear flows on the torus \mathbb{T}^d, d\geq 2 . We show that if the vector of the translation flow is Liouvillian (i.e. well approximated by rationals), then for a residual set of time change functions in the C^\infty topology, the reparameterized flow is weak mixing. If this is not the case, i.e. if the vector of the linear flow is Diophantine, it follows from a result of Kolmogorov on the two torus, and its generalization to any dimension by Herman, that any C^\infty reparameterization of the flow is C^\infty conjugate to a linear flow. More generally, in any given class of differentiability C^r for the time change function \phi , we give the optimal arithmetical condition on the vector of the translation flow that guarantees the existence of a residual set in the C^r topology of weak mixing reparameterizations. In the real analytic case, the optimal arithmetical condition for the generic display of weak mixing under time change is also given. As a consequence of our results on reparameterizations of Liouvillian linear flows, we obtain that an aperiodic smooth flow on the two-dimensional torus is in general weak mixing. We also deduce the existence on the torus of analytic diffeomorphisms that are rank one and weak mixing.

Analytic uniquely ergodic volume preserving maps on odd spheres

Abstract.We consider the linear cocycle (T, A) induced by a measure preserving dynamical system T : X → X and a map A: X → SL(2, ℝ). We address the dependence of the upper Lyapunov exponent of (T, A) on the dynamics T when the map A is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero.In the weak topology we deduce that if X is a compact connected manifold, then for a Cr (r ≥ 1) open and dense set of maps A, either (T, A) is uniformly hyperbolic for every T, or the Lyapunov exponents of (T, A) vanish for the generic measurable T.For the continuous case, we obtain that if X is of dimension greater than 2, then for a Cr (r ≥ 1) generic map A, there is a residual set of volume-preserving homeomorphisms T for which either (T, A) is uniformly hyperbolic or the Lyapunov exponents of (T, A) vanish.

KAM-tori near an analytic elliptic fixed point

We show that a stably ergodic diffeomorphism can be C 1 approximated by a diffeomorphism having stably non-zero Lyapunov

On the ergodicity of the Weyl sums cocycle

We construct, over some minimal translations of the two torus, special flows under a differentiable ceiling function that combine the properties of mixing and rank one.