Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Rare events, escape rates and quasistationarity: some exact formulae

We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given

On the recurrence and robust properties of Lorenz'63 model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.Comment: 44 pages, 7 figures, revised version accepted for pubblicatio

Acquired resistance to oxaliplatin is not directly associated with increased resistance to DNA damage in SK-N-ASrOXALI4000, a newly established oxaliplatin-resistant sub-line of the neuroblastoma cell line SK-N-AS

The formation of acquired drug resistance is a major reason for the failure of anti-cancer therapies after initial response. Here, we introduce a novel model of acquired oxaliplatin resistance, a sub-line of the non-MYCN-amplified neuroblastoma cell line SK-N-AS that was adapted to growth in the presence of 4000 ng/mL oxaliplatin (SK-N-ASrOXALI4000). SK-N-ASrOXALI4000 cells displayed enhanced chromosomal aberrations compared to SK-N-AS, as indicated by 24-chromosome fluorescence in situ hybridisation. Moreover, SK-N-ASrOXALI4000 cells were resistant not only to oxaliplatin but also to the two other commonly used anti-cancer platinum agents cisplatin and carboplatin. SK-N-ASrOXALI4000 cells exhibited a stable resistance phenotype that was not affected by culturing the cells for 10 weeks in the absence of oxaliplatin. Interestingly, SK-N-ASrOXALI4000 cells showed no cross resistance to gemcitabine and increased sensitivity to doxorubicin and UVC radiation, alternative treatments that like platinum drugs target DNA integrity. Notably, UVC-induced DNA damage is thought to be predominantly repaired by nucleotide excision repair and nucleotide excision repair has been described as the main oxaliplatin-induced DNA damage repair system. SK-N-ASrOXALI4000 cells were also more sensitive to lysis by influenza A virus, a candidate for oncolytic therapy, than SK-N-AS cells. In conclusion, we introduce a novel oxaliplatin resistance model. The oxaliplatin resistance mechanisms in SK-N-ASrOXALI4000 cells appear to be complex and not to directly depend on enhanced DNA repair capacity. Models of oxaliplatin resistance are of particular relevance since research on platinum drugs has so far predominantly focused on cisplatin and carboplatin

Mixing rates and limit theorems for random intermittent maps

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps $T_\alpha$ using the full parameter range $0< \alpha < \infty$, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest $\alpha$) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the \emph{quenched} dynamics. The main applications of this estimate are to \textit{limit laws} (in particular, CLT and stable laws, depending on the parameters chosen in the range $0 < \alpha < 1$) for the associated skew product; these are detailed in Theorem 3.2. Since our estimates in Theorem 1.1 also hold for $1 \leq \alpha < \infty$ we study a piecewise affine version of our random transformations, prove existence of an infinite ($\sigma-$finite) invariant measure and study the corresponding correlation asymptotics. To the best of our knowledge, this latter kind of result is completely new in the setting of random transformations.Comment: After publication, a silly mistake was found in definition of the skew product (4.1). The updated version includes a correction of equation (4.1) in the published version. All results of the paper are unaffected. In particular, everything in sections 1-3 and the proof in section 4 are unchanged from the published versio