On the complexity of some birational transformations

Using three different approaches, we analyze the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach consists in the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis, and the third method is more numerical, using integer arithmetics. Each method has its own domain of application, but they give corroborating results, and lead us to a conjecture on the complexity of a class of maps constructed from matrix inversions

Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit

We investigate quantised maps of the torus whose classical analogues are ergodic but not mixing. Their quantum spectral statistics shows non-generic behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the map to a spin 1/2, which corresponds to changing the quantisation without altering the classical limit of the dynamics on the torus, we numerically observe a transition to RMT statistics. The results are interpreted in terms of semiclassical trace formulae for the maps with and without spin respectively. We thus have constructed quantum systems with non-mixing classical limit which show generic (i.e. RMT) spectral statistics. We also discuss the analogous situation for an almost integrable map, where we compare to Semi-Poissonian statistics.Comment: 29 pages, 20 figure

Ruelle-Perron-Frobenius spectrum for Anosov maps

We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows to obtain easily very strong spectral stability results, which in turn imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe

Integrate and Fire Neural Networks, Piecewise Contractive Maps and Limit Cycles

We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincar\'e map associated to the system. We show that for strong interactions the Poincar\'e map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincar\'e map under study, but also to a wide class of general n-dimensional piecewise contractive maps.Comment: 46 pages. In this version we added many comments suggested by the referees all along the paper, we changed the introduction and the section containing the conclusions. The final version will appear in Journal of Mathematical Biology of SPRINGER and will be available at http://www.springerlink.com/content/0303-681

Statistical and dynamical properties of covariant lyapunov vectors in a coupled atmosphere-ocean model—multiscale effects, geometric degeneracy, and error dynamics

We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. There are two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. Interestingly, the tangent space of the unstable manifold has substantial projection on both atmospheric and oceanic components. The results show the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. They also suggest the need for an accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic

Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.Comment: 34 pages, 7 figures; Journal of Statistical Physics 201

The histone deacetylase inhibitor SAHA acts in synergism with fenretinide and doxorubicin to control growth of rhabdoid tumor cells

Background: Rhabdoid tumors are highly aggressive malignancies affecting infants and very young children. In many instances these tumors are resistant to conventional type chemotherapy necessitating alternative approaches. Methods: Proliferation assays (MTT), apoptosis (propidium iodide/annexin V) and cell cycle analysis (DAPI), RNA expression microarrays and western blots were used to identify synergism of the HDAC (histone deacetylase) inhibitor SAHA with fenretinide, tamoxifen and doxorubicin in rhabdoidtumor cell lines. Results: HDAC1 and HDAC2 are overexpressed in primary rhabdoid tumors and rhabdoid tumor cell lines. Targeting HDACs in rhabdoid tumors induces cell cycle arrest and apoptosis. On the other hand HDAC inhibition induces deregulated gene programs (MYCC-, RB program and the stem cell program) in rhabdoid tumors. These programs are in general associated with cell cycle progression. Targeting these activated pro-proliferative genes by combined approaches of HDAC-inhibitors plus fenretinide, which inhibits cyclinD1, exhibit strong synergistic effects on induction of apoptosis. Furthermore, HDAC inhibition sensitizes rhabdoid tumor cell lines to cell death induced by chemotherapy. Conclusion: Our data demonstrate that HDAC inhibitor treatment in combination with fenretinide or conventional chemotherapy is a promising tool for the treatment of chemoresistant rhabdoid tumors.<br

Complete spectral data for analytic Anosov maps of the torus

Using analytic properties of Blaschke factors we construct a family of analytic hyperbolic diffeomorphisms of the torus for which the spectral properties of the associated transfer operator acting on a suitable Hilbert space can be computed explicitly. As a result, we obtain explicit expressions for the decay of correlations of analytic observables without resorting to any kind of perturbation argument.Comment: 19 pages, 4 figure

ATRT–SHH comprises three molecular subgroups with characteristic clinical and histopathological features and prognostic significance

Atypical teratoid/rhabdoid tumor (ATRT) is an aggressive central nervous system tumor characterized by loss of SMARCB1/INI1 protein expression and comprises three distinct molecular groups, ATRT–TYR, ATRT–MYC and ATRT–SHH. ATRT–SHH represents the largest molecular group and is heterogeneous with regard to age, tumor location and epigenetic profile. We, therefore, aimed to investigate if heterogeneity within ATRT–SHH might also have biological and clinical importance. Consensus clustering of DNA methylation profiles and confirmatory t-SNE analysis of 65 ATRT–SHH yielded three robust molecular subgroups, i.e., SHH-1A, SHH-1B and SHH-2. These subgroups differed by median age of onset (SHH-1A: 18 months, SHH-1B: 107 months, SHH-2: 13 months) and tumor location (SHH-1A: 88% supratentorial; SHH-1B: 85% supratentorial; SHH-2: 93% infratentorial, often extending to the pineal region). Subgroups showed comparable SMARCB1 mutational profiles, but pathogenic/likely pathogenic SMARCB1 germline variants were over-represented in SHH-2 (63%) as compared to SHH-1A (20%) and SHH-1B (0%). Protein expression of proneural marker ASCL1 (enriched in SHH-1B) and glial markers OLIG2 and GFAP (absent in SHH-2) as well as global mRNA expression patterns differed, but all subgroups were characterized by overexpression of SHH as well as Notch pathway members. In a Drosophila model, knockdown of Snr1 (the fly homologue of SMARCB1) in hedgehog activated cells not only altered hedgehog signaling, but also caused aberrant Notch signaling and formation of tumor-like structures. Finally, on survival analysis, molecular subgroup and age of onset (but not ASCL1 staining status) were independently associated with overall survival, older patients (> 3 years) harboring SHH-1B experiencing relatively favorable outcome. In conclusion, ATRT–SHH comprises three subgroups characterized by SHH and Notch pathway activation, but divergent molecular and clinical features. Our data suggest that molecular subgrouping of ATRT–SHH has prognostic relevance and might aid to stratify patients within future clinical trials. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s00401-022-02424-5