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Bassam Fayad

Publications and source records attributed to Bassam Fayad.

52 records · Page 3Linked to original sources
arXiv2012

On manifolds supporting distributionally uniquely ergodic diffeomorphisms

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].

math.DS↗
arXiv2010

A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies

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

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arXiv2006

Nonstandard Smooth Realizations of Liouville Rotations

We augment the method of $C^\infty$ conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

math.DS↗
arXiv2006

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

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 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 $C^r$ ($r \ge 1$) 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 $C^r$ ($r \ge 1$) 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.

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arXiv2006

On the ergodicity of the Weyl sums cocycle

For $θ\in [0,1]$, we consider the map $T_\a: \T^2 \to \T^2$ given by $T_θ(x,y)=(x+θ,y+2x+θ)$. The skew product $f_\a: \T^2 \times \C \to \T^2 \times \C$ given by $f_θ(x,y,z)=(T_θ(x,y),z+e^{2 πi y})$ generates the so called Weyl sums cocycle $a_\a(x,n) = \sum_{k=0}^{n-1} e^{2πi(k^2θ+kx)}$ since the $n^{\rm th}$ iterate of $f_\a$ writes as $f_\a^n(x,y,z)=(T_\a^n(x,y),z+e^{2πiy} a_\a(2x,n))$. In this note, we improve the study developed by Forrest in \cite{forrest2,forrest} around the density for $x \in \T$ of the complex sequence ${\{a_\a(x,n)\}}_{n\in \N}$, by proving the ergodicity of $f_θ$ for a class of numbers $\a$ that contains a residual set of positive Hausdorff dimension in $[0,1]$. The ergodicity of $f_\a$ implies the existence of a residual set of full Haar measure of $x \in \T$ for which the sequence ${\{a_\a(x,n) \}}_{n \in \N}$ is dense.

math.NT↗
arXiv2005

On the ergodicity of cylindrical transformations given by the logarithm

Given $\a \in [0,1]$ and $φ: \T \to \R$ measurable, the {\it cylindircal cascade} $S_{\a,φ}$ is the map from $\T \times \R$ to itself given by $S_{\a,φ} (x,y) = (x+\a,y+φ(x))$ that naturally appears in the study of some ordinary differential equations on $\R^3$. In this paper, we prove that for a set of full Lebesgue measure of $\a \in [0,1]$ the cylindrical cascades $S_{\a,φ}$ are ergodic for every smooth function $φ$ with a logarithmic singularity, provided that the average of $φ$ vanishes. Closely related to $S_{\a,φ}$ are the special flows constructed above $R_\a$ and under $φ+c$ where $c \in \R$ is such that $φ+c>0$. In the case of a function $φ$ with an asymmetric logarithmic singularity our result gives the first examples of ergodic cascades $S_{\a,φ}$ with the corresponding special flows being mixing. Indeed, when the latter flows are mixing the usual techniques used to prove the {\it essential value criterion} for $S_{\a,φ}$, that is equivalent to ergodicity, fail and we device a new method to prove this criterion that we hope could be useful in tackling other problems of ergodicity for cocycles preserving an infinite measure.

math.DS↗
arXiv2005

Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary

Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action ${\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t$, preserving a smooth volume $μ$. For any Liouville number $\a$ we construct a sequence of area-preserving diffeomorphisms $H_n$ such that the sequence $H_n\circ S_\a\circ H_n^{-1}$ converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc $\DD^2=\{x^2+y^2\leq 1\}$ or the closed annulus $\AAA=\TT\times [0,1]$ this result proves the following dichotomy: $\a \in \RR \setminus\QQ$ is Diophantine if and only if there is no ergodic diffeomorphism of $M$ whose rotation number on the boundary equals $α$ (on at least one of the boundaries in the case of $\AAA$). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if $\a$ is Diophantine, then any area preserving diffeomorphism with rotation number $\a$ on the boundary (on at least one of the boundaries in the case of $\AAA$) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.

math.DS↗
arXiv2005

Constructions in elliptic dynamics

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and the second author more than thirty years ago \cite{AK}. 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 in \cite{FH} and continuing with exotic invariant sets of rational maps of the Riemann sphere \cite{H3}, and the construction of invariant tori with nonstandard and unexpected behavior in the context of KAM theory \cite{H1, H2}. 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 possibility of mixing in the context of Liouvillean dynamics discovered by the first author \cite{F1, F2}.

math.DS↗
arXiv2005

Two remarks on the Shrinking Target Property

We show that a translation of vector V on the torus has the monotone shrinking target property if and only if the vector V is of constant type. Then, using reparametrizations of linear flows, we show that there exist area preserving real analytic maps of the three torus that are mixing of all orders and do not enjoy the monotone shrinking target property.

math.DS↗
arXiv2004

Smooth mixing flows with singular spectra

We give a geometric criterion that guaranteesa purely singular spectral type for a dynamical system on a Riemannian manifold. The criterion, that is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we use it to construct examples of smooth mixing flows on the three torus with purely singular spectra.

math.DS↗
arXiv2004

Rank one and mixing differentiable flows

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.

math.DS↗
arXiv2004

Continuous spectrum on laminations over Aubry-Mather sets

If we perturb a completely integrable Hamiltonian system with two degrees of freedom, the perturbed flow might display, on every energy level, invariant sets that are laminations over Aubry-Mather sets of a Poincaré section of the flow. Each one of these laminations carry a unique invariant probability measure for the flow on which mixing is impossible in this low dimensional frame. We prove that if the Aubry-Mather set has exactly one orbit of gaps and is hyperbolic then the special flow over it with any smooth ceiling function will be conjugate to a suspension with a constant ceiling function, failing hence to be weak mixing or even topologically weak mixing. To the contrary, if the Aubry-Mather set has more than one orbit of gaps with at least two in a general position then the special flow over it will in general be weak mixing.

math.DS↗