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

Promoting the 3Rs to enhance the OECD fish toxicity testing framework.

Fish toxicity testing has been conducted since the 1860's in order to help define safe levels of chemical contaminants in lakes, rivers and coastal waters. The historical emphasis on acute lethality testing of chemicals has more recently focussed on long term sublethal effects of chemicals on fish and their prey species. Fish toxicity testing is now embedded in much environment legislation on chemical safety while it is recognized that animal use should be Replaced, Reduced and Refined (the 3Rs) where possible. The OECD Fish Toxicity Testing Framework provides a useful structure with which to address the needs of environmental safety assessment whilst implementing the 3Rs. This commentary aims to promote the implementation of the recommendations of the OECD Fish Toxicity Testing Framework

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la

Robust entropy expansiveness implies generic domination

Let $f: M \to M$ be a $C^r$-diffeomorphism, $r\geq 1$, defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$, and let $H(p)$ be the homoclinic class associated to the hyperbolic periodic point $p$. We prove that if there exists a $C^1$ neighborhood $\mathcal{U}$ of $f$ such that for every $g\in {\mathcal U}$ the continuation $H(p_g)$ of $H(p)$ is entropy-expansive then there is a $Df$-invariant dominated splitting for $H(p)$ of the form $E\oplus F_1\oplus... \oplus F_c\oplus G$ where $E$ is contracting, $G$ is expanding and all $F_j$ are one dimensional and not hyperbolic.Comment: 24 page

Relative Decay of Fecal Indicator Bacteria and Human-Associated Markers: A Microcosm Study Simulating Wastewater Input into Seawater and Freshwater

Fecal contaminations of inland and coastal waters induce risks to human health and economic losses. To improve water management, specific markers have been developed to differentiate between sources of contamination. This study investigates the relative decay of fecal indicator bacteria (FIB, Escherichia coli and enterococci) and six human-associated markers (two bacterial markers: Bacteroidales HF183 (HF183) and Bifidobacterium adolescentis (BifAd); one viral marker: genogroup II F-specific RNA bacteriophages (FRNAPH II); three chemical markers: caffeine and two fecal stanol ratios) in freshwater and seawater microcosms seeded with human wastewater. These experiments were performed in darkness, at 20 °C and under aerobic conditions. The modeling of the decay curves allows us (i) to compare FIB and markers and (ii) to classify markers according to their persistence in seawater (FRNAPH II < HF183, stanol ratios < BifAd, caffeine) and in freshwater (HF183, stanol ratios < FRNAPH II < BifAd < caffeine). Although those results depend on the experimental conditions, this study represents a necessary step to develop and validate an interdisciplinary toolbox for the investigation of the sources of fecal contaminations

Infinitesimal Lyapunov functions for singular flows

We present an extension of the notion of infinitesimal Lyapunov function to singular flows, and from this technique we deduce a characterization of partial/sectional hyperbolic sets. In absence of singularities, we can also characterize uniform hyperbolicity. These conditions can be expressed using the space derivative DX of the vector field X together with a field of infinitesimal Lyapunov functions only, and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region.Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2) of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the statement of this item was false; corrected items (1) and (2) of Theorem 2.23 and their proofs. Included Example 6 on smooth reduction of families of quadratic forms. The published version in Math Z journal needs an errat

Cartographie de la végétation terrestre des îlots marins de la Réserve Naturelle des Bouches de Bonifacio

Rapport Office de l'environnement de la Corse/CNR