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• 1. Albeverio, Sergio
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On the Markovian Behavior of p-Adic Random Dynamical Systems2001In: Russ. J. Math. Phys., Vol. 8, no 2, p. 135-152Article in journal (Refereed)
• 2.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Applied Algebraic Dynamics2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 4, p. 360-362Article in journal (Refereed)
• 3.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Divergence and convergence of conjugacies in non-Archimedean dynamics2010In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 508, p. 89-109Article in journal (Refereed)

We continue the study of the linearizability near an indifferent fixed point of a power series f, defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz in 1981 that Siegel’s linearization theorem is true also for non-Archimedean fields. However, they also showed that the condition in Siegel’s theorem is ‘usually’ not satisfied over fields of prime characteristic. Indeed, as proven by the author in a former paper, there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analytically linearizable. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges.

• 4.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Estimates of linearization discs in p-adic dynamics with application to ergodicityManuscript (preprint) (Other academic)
• 5.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Linearization in Ultrametric Dynamics in Fields of Characteristic Zero — Equal Characteristic Case2009In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 4, p. 307-316Article in journal (Refereed)

Let K be a complete ultrametric field of characteristic zero whose corresponding residue field k is also of characteristic 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 examples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaining 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 [5, 18, 35, 42] later generalized in [28, 30], and in [29, 31] concerning the prime characteristic case.

• 6.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On Siegel's linearization theorem for fields of prime characteristic2004In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 17, no 3, p. 745-763Article in journal (Refereed)
• 7.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point2007Doctoral thesis, monograph (Other academic)

We consider the problem of local linearization of power series defined over complete valued fields. The complex field case has been studied since the end of the nineteenth century, and renders a delicate number theoretical problem of small divisors related to diophantine approximation. Since a work of Herman and Yoccoz in 1981, there has been an increasing interest in generalizations to other valued fields like p-adic fields and various function fields. We present some new results in this domain of research. In particular, for fields of prime characteristic, the problem leads to a combinatorial problem of seemingly great complexity, albeit of another nature than in the complex field case.

In cases for which linearization is possible, we estimate the size of linearization discs and prove existence of periodic points on the boundary. We also prove that transitivity and ergodicity is preserved under the linearization. In particular, transitivity and ergodicity on a sphere inside a non-Archimedean linearization disc is possible only for fields of p-adic numbers.

• 8.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Some results on convergence of conjugating functions over non-Archimedean fields2002In: Dynamical systems from number theory to physics - 2, 2002, p. 67-77Conference paper (Refereed)
• 9.
The size of quadratic p-adic linearization disks2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 248, p. 872-894Article in journal (Refereed)

We find the exact radius of linearization disks at indifferentfixed points ofquadratic maps in Cp. We also show thatthe radius is invariant under power series perturbations.Localizing all periodic orbits of these quadratic-like maps wethen show that periodic points are not the only obstruction for linearization. In so doing, we provide the first known examples in the dynamics ofpolynomials over Cp where the boundary of the linearization disk does not contain any periodic point.

• 10.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering. Matematik.
Uniqueness for p-adic meromorphic products2006In: Mathematical ReviewsArticle, book review (Other (popular science, discussion, etc.))

MR2232636

Boussaf, Kamal(F-CLEF2-LPM)

Uniqueness for $p$-adic meromorphic products. (English summary)

Bull. Belg. Math. Soc. Simon Stevin 9 (2002), suppl., 11--23.

32P05 (32H04)

PDF Doc Del Clipboard Journal Article Make Link

In the paper under review the author looks for bi-unique range sets (bi-urs) for the family of unbounded meromorphic products on an open disk. More precisely, let $K$ be a complete ultrametric algebraically closed field of characteristic zero, and let $\scr{M}(K)$ be the field of meromorphic functions in $K$. Denote by $\scr{MP}_{u}(K,R)$ the subset of meromorphic products admitting an irreducible form $\prod_{n=0}^{\infty}\frac{x-a_n}{x-b_n}$ such that $\prod_{n=0, b_n\neq 0}^{\infty}\frac{|b_n|}{R}=0$. The main result in the paper under review implies that for every $n\geq5$, there exist sets $S$ of $n$ elements in $K$ such that $(S,\{\infty\})$ is a bi-urs for $\scr{MP}_u(K,R)$.

Earlier, A. Boutabaa and A. Escassut proved that for every $n\geq5$, there exist sets $S$ of $n$ elements in $K$ such that $(S,\{w\})$ is a bi-urs for $\scr{M}(K)$. H. H. Khoi and T. T. H. An showed the existence of bi-urs for $\scr{M}(K)$ of the form $(\{a_1,a_2,a_3,a_4\},\{\infty\})$.

Reviewed by Karl-Olof Lindahl

• 11.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
Växjö University, Faculty of Mathematics/Science/Technology, School of Mathematics and Systems Engineering.
On Ergodic Behavior of p-adic dynamical systems2001In: Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 4, no 4, p. 569-577Article in journal (Refereed)
• 12.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Geometric location of periodic points of 2-ramified power series2018In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 465, no 2, p. 762-794Article in journal (Refereed)

In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic p. More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period pn of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the pnth iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.

• 13.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Generic parabolic points are isolated in positive characteristic2016In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 29, no 5, p. 1596-1621Article in journal (Refereed)

We study germs in one variable having a parabolic fixed point at the origin, over an ultrametric ground field of positive characteristic. It is conjectured that for such a germ the origin is isolated as a periodic point. Our main result is an affirmative solution of this conjecture in the case of a generic germ with a prescribed multiplier. The genericity condition is explicit: That the power series is minimally ramified, i.e., that the degree of the first nonlinear term of each of its iterates is as small as possible. Our main technical result is a computation of the first significant terms of a minimally ramified power series. From this we obtain a lower bound for the norm of nonzero periodic points, from which we deduce our main result. As a by-product we give a new and self-contained proof of a characterization of minimally ramified power series in terms of the iterative residue.

• 14.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Pontífica Univerisdád Católica de Santiago de Chile, Chile.
Optimal cycles in ultrametric dynamics and minimally ramified power series2016In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 152, no 01, p. 187-222Article in journal (Refereed)

We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are "optimal" in the following sense: They minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. In those cases we exhibit a higher degree polynomial such that all of its cycles are optimal. The proof of these results reveals a connection between the geometric location of periodic points of ultrametric power series and the lower ramification numbers of wildly ramified field automorphisms. We also give an extension of Sen's theorem on wildly ramified field automorphisms that is of independent interest.

• 15.
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
Michigan State University.
On Hyperbolic Fixed Points in Ultrametric Dynamics2010In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 3, p. 232-240Article in journal (Refereed)

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.

• 16.
Linnaeus University, Faculty of Technology, Department of Mathematics.
Linnaeus University, Faculty of Technology, Department of Mathematics. Linnaeus University, Faculty of Technology, Department of Mathematics.
Distributional Reinforcement Learning with EnsemblesManuscript (preprint) (Other academic)

It is well-known that ensemble methods often provide enhanced performance in reinforcement learning. In this paper we explore this concept further by using group-aided training within the distributional reinforcement learning paradigm. Specifically, we propose an extension to categorical reinforcement learning, where distributional learning targets are implicitly based on the total information gathered by an ensemble. We empirically show that this may lead to much more robust initial learning, a stronger individual performance level and good efficiency on a per-sample basis.

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