An Isotopic Perturbation Lemma Along Periodic Orbits

A well-known lemma by John Franks asserts that one obtains any perturbation of the derivative of a diffeomorphism along a periodic orbit by a $C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of the orbit. However, one does not control where the invariant manifolds of the orbit are, after perturbation. We show that if the perturbated derivative is obtained by an isotopy along which some strong stable/unstable manifolds of some dimensions exist, then the Franks perturbation can be done preserving the corresponding stable/unstable semi-local manifolds. This is a general perturbative tool in $C^1$-dynamics that has many consequences. We give simple examples of such consequences, for instance a generic dichotomy between dominated splitting and small stable/unstable angles inside homoclinic classes.Comment: 51 pages. arXiv admin note: substantial text overlap with arXiv:0912.112

Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems $\xi_{t+1} = A(u_t) \cdot \xi_t$, where the states $\xi_t$ are in projective space $\mathbb{R}P^{d-1}$, inputs $u_t$ are in a manifold $U$ of arbitrary finite dimension, and $A \colon U \to GL(d,\mathbb{R})$ is a differentiable mapping. An input sequence $(u_0,\ldots,u_{N-1})$ is called universally regular if for any initial state $\xi_0 \in \mathbb{R}P^{d-1}$, the derivative of the time-$N$ state with respect to the inputs is onto. In this paper we deal with the universal regularity of constant input sequences $(u_0, \dots, u_0)$. Our main result states that generically in the space of such systems, for sufficiently large $N$, all constant inputs of length $N$ are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a $C^2$-open and $C^\infty$-dense set of maps $A$, and $N$ only depends on $d$ and on the dimension of $U$. We also show that the inputs on that discrete set are nearly universally regular; indeed there is a unique non-regular initial state, and its corank is $1$. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.Comment: 48 pages. This version incorporates suggestions and corrections by the referees. It also includes arXiv:1201.2217 as an Appendi

Rotation numbers of perturbations of smooth dynamics

The small perturbations of a linear cocycle have a relative rotation number associated to each pair formed of an invariant measure of the base dynamics and a $2$-dimensional bundle of the finest dominated splitting (provided that some orientation is preserved). The properties of that relative rotation number allow some small steps towards dichotomies between complex eigenvalues and dominated splittings in higher dimensions and higher regularity.Comment: 30 page

Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

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 which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor set.Comment: Final version. To appear in Duke Mathematical Journa

Occurrence of Bacterial Pathogens and Human Noroviruses in Shellfish-Harvesting Areas and Their Catchments in France

During a 2-year study, the presence of human pathogenic bacteria and noroviruses was investigated in shellfish, seawater and/or surface sediments collected from three French coastal shellfish-harvesting areas as well as in freshwaters from the corresponding upstream catchments. Bacteria isolated from these samples were further analyzed. Escherichia coli isolates classified into the phylogenetic groups B2, or D and enterococci from Enterococcus faecalis and E. faecium species were tested for the presence of virulence genes and for antimicrobial susceptibility. Salmonella members were serotyped and the most abundant serovars (Typhimurium and its monophasic variants and Mbandaka) were genetically characterized by high discriminative subtyping methods. Campylobacter and Vibrio were identified at the species level, and haemolysin-producing Vibrio parahaemolyticus were searched by tdh- and trh- gene detection. Main results showed a low prevalence of Salmonella in shellfish samples where only members of S. Mbandaka were found. Campylobacter were more frequently isolated than Salmonella and a different distribution of Campylobacter species was observed in shellfish compared to rivers, strongly suggesting possible additional inputs of bacteria. Statistical associations between enteric bacteria, human noroviruses (HuNoVs) and concentration of fecal indicator bacteria revealed that the presence of Salmonella was correlated with that of Campylobacter jejuni and/or C. coli as well as to E. coli concentration. A positive correlation was also found between the presence of C. lari and the detection of HuNoVs. This study highlights the importance of simultaneous detection and characterization of enteric and marine pathogenic bacteria and human noroviruses not only in shellfish but also in catchment waters for a hazard assessment associated with microbial contamination of shellfish

Dynamic instability in absence of dominated splittings.

On veut comprendre les implications dynamiques de l'absence de décompositions dominées. Une décomposition dominée est une forme affaiblie d'hyperbolicité où l'espace tangent d'une variété est la somme directe de sous-fibrés invariants, rangés du plus contracté au plus dilaté par la dynamique. On commence par répondre à une ancienne question de Hirsch, Pugh et Shub, en démontrant l'existence de métriques adaptées pour les décompositions dominées. Sur les surfaces, Mañé a démontré une dichotomie $C^1$-générique entre hyperbolicité et phénomènes de Newhouse (une infinité de puits/sources). Pour cela, il a prouvé que lorsque les décompositions dominées le long d'une orbite périodique sont trop faibles, une $C^1$-pertubation crée un puits ou une source. On généralise ce dernier énoncé à toute dimension en se ramenant à l'étude de cocycles linéaires, grâce à un lemme de Franks. Abdenur, Bonatti et Crovisier en ont déduit des dichotomies $C^1$-génériques en toute dimension entre phénomènes de Newhouse et décompositions dominées sur les ensembles non-errants. Les deux derniers chapitres sont consacrés à la création de tangences homoclines en l'absence de décomposition dominée stable/instable, dans le prolongement de travaux de Wen. Enfin, dans le dernier chapitre, on montre que si la classe homocline d'une selle $P$ n'a pas de décomposition dominée de même indice que $P$, une perturbation crée une tangence associée à $P$.We want to understand the dynamics in absence of dominated splittings. A dominated splitting is a weak form of hyperbolicity where the tangent bundle splits into invariant subbundles, each of them is more contracted or less expanded by the dynamics than the next one. We first answer an old question from Hirsch, Pugh and Shub, and show the existence of adapted metrics for dominated splittings.Mañé found on surfaces a $C^1$-generic dichotomy between hyperbolicity and Newhouse phenomenons (infinitely many sinks/sources). For that purpose, he showed that without a strong enough dominated splitting along one periodic orbit, a $C^1$-perturbation creates a sink or a source. We generalise that last statement to any dimension, reducing our study to linear cocycles thanks to a Franks' lemma. Abdenur, Bonatti and Crovisier then deduced $C^1$-generic dichotomies in any dimension between Newhouse phenomenons and dominated splittings on the non-wandering sets. The last two chapters are dedicated to creating homoclinic tangencies in absence of stable/unstable dominated splittings, thus extending results of Wen. In the last chapter we finally get that if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a perturbation creates a tangency associated to $P$

Instabilité de la dynamique en l'absence de décompositions dominées.

We want to understand the dynamics in absence of dominated splittings. A dominated splitting is a weak form of hyperbolicity where the tangent bundle splits into invariant subbundles, each of them is more contracted or less expanded by the dynamics than the next one. We first answer an old question from Hirsch, Pugh and Shub, and show the existence of adapted metrics for dominated splittings.Mañé found on surfaces a $C^1$-generic dichotomy between hyperbolicity and Newhouse phenomenons (infinitely many sinks/sources). For that purpose, he showed that without a strong enough dominated splitting along one periodic orbit, a $C^1$-perturbation creates a sink or a source. We generalise that last statement to any dimension, reducing our study to linear cocycles thanks to a Franks' lemma. Abdenur, Bonatti and Crovisier then deduced $C^1$-generic dichotomies in any dimension between Newhouse phenomenons and dominated splittings on the non-wandering sets. The last two chapters are dedicated to creating homoclinic tangencies in absence of stable/unstable dominated splittings, thus extending results of Wen. In the last chapter we finally get that if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a perturbation creates a tangency associated to $P$.On veut comprendre les implications dynamiques de l'absence de décompositions dominées. Une décomposition dominée est une forme affaiblie d'hyperbolicité où l'espace tangent d'une variété est la somme directe de sous-fibrés invariants, rangés du plus contracté au plus dilaté par la dynamique. On commence par répondre à une ancienne question de Hirsch, Pugh et Shub, en démontrant l'existence de métriques adaptées pour les décompositions dominées. Sur les surfaces, Mañé a démontré une dichotomie $C^1$-générique entre hyperbolicité et phénomènes de Newhouse (une infinité de puits/sources). Pour cela, il a prouvé que lorsque les décompositions dominées le long d'une orbite périodique sont trop faibles, une $C^1$-pertubation crée un puits ou une source. On généralise ce dernier énoncé à toute dimension en se ramenant à l'étude de cocycles linéaires, grâce à un lemme de Franks. Abdenur, Bonatti et Crovisier en ont déduit des dichotomies $C^1$-génériques en toute dimension entre phénomènes de Newhouse et décompositions dominées sur les ensembles non-errants. Les deux derniers chapitres sont consacrés à la création de tangences homoclines en l'absence de décomposition dominée stable/instable, dans le prolongement de travaux de Wen. Enfin, dans le dernier chapitre, on montre que si la classe homocline d'une selle $P$ n'a pas de décomposition dominée de même indice que $P$, une perturbation crée une tangence associée à $P$

Generation of homoclinic tangencies by C1C^1-perturbations.

Given a [C^1] -diffeomorphism [f] of a compact manifold, we show that if the stable/unstable dominated splitting along a saddle is weak enough, then there is a small [C^1] -perturbation that preserves the orbit of the saddle and that generates a homoclinic tangency related to it. Moreover, we show that the perturbation can be performed preserving a homoclinic relation to another saddle. We derive some consequences on homoclinic classes. In particular, if the homoclinic class of a saddle [P] has no dominated splitting of same index as [P] , then a [C^1] -perturbation generates a homoclinic tangency related to [P]