Distribution of periodic points of polynomial diffeomorphisms of C^2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of \C^2: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$

Comparative aspects of phytase and xylanase effects on performance, mineral digestibility, and ileal phytate degradation in broilers and turkeys

Two experiments were performed, using broilers or turkeys, each utilizing a 3 × 2 factorial arrangement, to compare their response to phytase and xylanase supplementation with growth performance, nutrient digestibility, and ileal phytate degradation as response criteria. For both experiments, 960 Ross 308 or 960 BUT 10 (0-day-old) birds were allocated to 6 treatments: (1) control diet, containing phytase at 500 FTU/kg; (2) the control diet with xylanase (16,000 BXU/kg); (3) the control diet supplemented on top with phytase (1,500 FTU/kg); (4) diet supplemented with 1,500 FTU/kg phytase and xylanase (16,000 BXU/kg); (5) the control diet supplemented with phytase (3,000 FTU/kg); and (6) diet supplemented with 3,000 FTU/kg phytase and xylanase (16,000 BXU/kg). Each treatment had 8 replicates of 20 birds each. Water and diets based on wheat, soybean meal, oilseed rape meal, and barley were available ad libitum. Body weight gain and feed intake were measured from 0 to 28 D, and feed conversion ratio (FCR) corrected for mortality was calculated. Ileal digestibility for dry matter and minerals on day 7 and 28 were analyzed in addition to levels of inositol phosphate esters (InsP6-3) and myo-inositol. Statistical comparisons were performed using ANOVA. Xylanase supplementation improved 28D FCR in broilers and turkeys. Increasing doses of phytase reduced FI and improved FCR only in broilers. In broilers, the age × phytase interaction for phosphorous digestibility showed that increasing phytase dose was more visible on day 7, than on day 28. Mineral digestibility was lower in 28-day-old turkey compared with 7-day-old turkey. InsP6 disappearance increased with increasing phytase levels in both species, with lower levels analyzed in turkeys. InsP6 disappearance was greater in younger turkeys (day 7 compared with day 28). In conclusion, although broilers and turkeys shared several similarities in their growth and nutrient utilization responses, the outcomes of the 2 trials also differed in many aspects. Whether this is because of difference in diets (InsP or Ca level) or differences between species needs further investigation

Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems

Practice pointer: Using the new UK-WHO growth charts

The new UK growth charts for children aged 0-4 years (designed using data from the new WHO standards) describe the optimal pattern of growth for all children, rather than the prevailing pattern in the UK (as with previous charts). The new charts are suitable for all ethnic groups and set breast feeding as the norm. UK children match the new charts well for length and height, but after age 6 months fewer children will be below the 2nd centile for weight or show weight faltering, and more will be above the 98th centile. The new charts look different: they have a separate preterm section, no lines between 0 and 2 weeks, and the 50th percentile is no longer emphasised. The charts give clear instructions on gestational correction, and there is a new chart for infants born before 32 weeks’ gestation. The instructions advise on when and how to measure and when a measurement or growth pattern is outside the normal range. The charts include a “look-up” tool for determining the body mass index centile from height and weight centiles without calculation and aid for predicting adult height. The charts and supporting educational materials can be downloaded from www.growthcharts.rcpch.ac.u

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.Comment: 21 pages; conjecture 2.8 was removed in proof; to appear in J. Geom. Ana

Maternal Risk Factors for Small-for-Gestational-Age Newborns in Mexico: Analysis of a Nationwide Representative Cohort

Background: Small for gestational age (SGA) is a key contributor to premature deaths and long-term complications in life. Improved characterization of maternal risk factors associated with this adverse outcome is needed to inform the development of interventions, track progress, and reduce the disease burden. This study aimed to identify socioeconomic, demographic, and clinical factors associated with SGA in Mexico. Methods: We analyzed administrative data from 1,841,477 singletons collected by the National Information Subsystem of Livebirths during 2017. Small-for-gestational-age was defined as being 12 years (aOR 0.63; 95% CI:0.6,0.66) compared with those without education. SGA was particularly likely to occur among primiparous (aOR 1.42; 95% CI: 1.39, 1.43), mothers living in very high deprivation localities (aOR 1.39; 95% CI: 1.36, 1.43), young (aOR 1.04; 95% CI: 1.02, 1.06), advanced age (aOR 1.14; 95% CI 1.09, 1.19), and mothers living in areas above 2,000 m (aOR 1.69; 95% CI: 1.65, 1.73). Antenatal care was associated with a reduced risk of SGA by 30% (aOR 0.7; 95% CI:0.67,0.73), 23% (OR 0.77; 95% CI:0.74,0.8), and 21% (OR 0.79; 95% CI:0.75,0.83), compared with those mothers who never received antenatal care, when women visited the clinic at the first, second and third trimester, respectively. Conclusion: Almost 7% of live births were found to be SGA. Parity, maternal age, education, place of residence, and social deprivation were significantly associated with this outcome. Antenatal care was protective. These findings imply that interventions focusing on early and adequate contact with health care facilities, reproductive health counseling, and maternal education should reduce SGA in Mexico

Failure to learn from feedback underlies word learning difficulties in toddlers at risk for autism

Children’s assignment of novel words to nameless objects, over objects whose names they know (mutual exclusivity; ME) has been described as a driving force for vocabulary acquisition. Despite their ability to use ME to fast-map words (Preissler & Carey, 2005), children with autism show impaired language acquisition. We aimed to address this puzzle by building on studies showing that correct referent selection using ME does not lead to word learning unless ostensive feedback is provided on the child’s object choice (Horst & Samuelson, 2008). We found that although toddlers aged 2;0 at risk for autism can use ME to choose the correct referent of a word, they do not benefit from feedback for long-term retention of the word–object mapping. Further, their difficulty using feedback is associated with their smaller receptive vocabularies. We propose that difficulties learning from social feedback, not lexical principles, limits vocabulary building during development in children at risk for autism

The role of insulin receptor substrate 2 in hypothalamic and β cell function

Insulin receptor substrate 2 (Irs2) plays complex roles in energy homeostasis. We generated mice lacking Irs2 in β cells and a population of hypothalamic neurons (RIPCreIrs2KO), in all neurons (NesCreIrs2KO), and in proopiomelanocortin neurons (POMCCreIrs2KO) to determine the role of Irs2 in the CNS and β cell. RIPCreIrs2KO mice displayed impaired glucose tolerance and reduced β cell mass. Overt diabetes did not ensue, because β cells escaping Cre-mediated recombination progressively populated islets. RIPCreIrs2KO and NesCreIrs2KO mice displayed hyperphagia, obesity, and increased body length, which suggests altered melanocortin action. POMCCreIrs2KO mice did not display this phenotype. RIPCreIrs2KO and NesCreIrs2KO mice retained leptin sensitivity, which suggests that CNS Irs2 pathways are not required for leptin action. NesCreIrs2KO and POMCCreIrs2KO mice did not display reduced β cell mass, but NesCreIrs2KO mice displayed mild abnormalities of glucose homeostasis. RIPCre neurons did not express POMC or neuropeptide Y. Insulin and a melanocortin agonist depolarized RIPCre neurons, whereas leptin was ineffective. Insulin hyperpolarized and leptin depolarized POMC neurons. Our findings demonstrate a critical role for IRS2 in β cell and hypothalamic function and provide insights into the role of RIPCre neurons, a distinct hypothalamic neuronal population, in growth and energy homeostasis