Ergodicity criteria for non-expanding transformations of 2-adic spheres

In the paper, we obtain necessary and sufficient conditions for ergodicity (with respect to the normalized Haar measure) of discrete dynamical systems $$ on 2-adic spheres $\mathbf S_{2^{-r}}(a)$ of radius $2^{-r}$, $r\ge 1$, centered at some point $a$ from the ultrametric space of 2-adic integers $\mathbb Z_2$. The map $f\colon\mathbb Z_2\to\mathbb Z_2$ is assumed to be non-expanding and measure-preserving; that is, $f$ satisfies a Lipschitz condition with a constant 1 with respect to the 2-adic metric, and $f$ preserves a natural probability measure on $\mathbb Z_2$, the Haar measure $\mu_2$ on $\mathbb Z_2$ which is normalized so that $\mu_2(\mathbb Z_2)=1$

On hyperbolic fixed points in ultrametric dynamics

Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.Comment: http://www.springerlink.com/content/?k=doi%3a%2810.1134%2fS2070046610030052%2

Calidad del agua del estero el sauce, valparaíso, chile central water quality in the el sauce estuary, valparaíso, central Chile

Indexación: Scopus.The main objective of this work was to evaluate the water quality of the El Sauce estuary and its tributaries. The El Sauce estuary basin is located in the town of Laguna Verde, Valparaíso, Central Chile. Sampling took place in the summer season of 2013 and 2015, in 11 stations located along the basin, five of them distributed from its origin to its mouth in the sea and six located before entering its tributaries. Point and non-point sources downloaded in its course were identified. The direct discharge of water from a sewage treatment plant in the area of origin of the estuary, and in its middle zone the percolation of a municipal landfill, stand out for their volume. Its mouth is affected by non-point sources of domestic waters in the town of Laguna Verde. The results show that the estuary is a shallow water course, which quality Class 4 (poor) in most of its extension presents due to the content of organic matter, nutrients, chlorides, and fecal contamination, not complying with environmental regulations for any use. There is a lack of management and control plans in the use of this important resource. It has become a risk to the community, who use the water of the stream both to irrigate subsistence agriculture and for recreation with a direct contact at its mouth.https://www.revistascca.unam.mx/rica/index.php/rica/article/view/RICA.534

Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case

Let $K$ be a complete ultrametric field of charactersitic zero whose corresponding residue field $\Bbbk$ is also of charactersitic zero. We give lower and upper bounds for the size of linearization disks for power series over $K$ near an indifferent fixed point. These estimates are maximal in the sense that there exist exemples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaning cases, i.e. the cases in which $K$ is either a $p$-adic field or a field of prime characteristic, were obtained in various papers on the $p$-adic case (Ben-Menahem:1988,Thiran/EtAL:1989,Pettigrew/Roberts/Vivaldi:2001,Khrennikov:2001) later generalized in (Lindahl:2009 arXiv:0910.3312), and in (Lindahl:2004 http://iopscience.iop.org/0951-7715/17/3/001/,Lindahl:2010Contemp. Math) concerning the prime characteristic case

Large derivatives, backward contraction and invariant densities for interval maps

Abstract. In this paper, we study the dynamics of a smooth multi-modal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |(fn)′(f(c)) | → ∞ as n→ ∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability µ which is absolutely continuous with respect to the Lebesgue measure and the density of µ belongs to Lp for all p < `max/(`max−1), where `max denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the deriv-atives along the critical orbits imply stronger backward contraction. 1

The distribution of galois orbits of points of small height in toric varieties

We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor D on a proper toric variety X over a global field K is monocritical if and only if for every generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of X. The key ingredient is the study of the v-adic modulus distribution of Galois orbits of generic D-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a closed subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.Burgos Gil was partially supported by the MINECO research projects MTM2013-42135-P,MTM2016-79400-P and ICMAT Severo Ochoa SEV-2015-0554. Philippon was partially supportedby the CNRS research project PICS 6381 “G ́eom ́etrie diophantienne et calcul formel” and theANR research project “Hauteurs, modularit ́e, transcendance”. Burgos Gil and Philippon werealso partially supported by the FP7-MC-IRSES project n◦612534 “MODULI”. Rivera-Letelierwas partially supported by FONDECYT grant 1141091 and NSF grant DMS-1700291. Sombra waspartially supported by the MINECO research projects MTM2012-38122-C03-02 and MTM2015-65361-P and through the “Mar ́ıa de Maeztu” program for units of excellence MDM-2014-044