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Lindahl, K.-O. & Nordqvist, J. (2018). Geometric location of periodic points of 2-ramified power series. Journal of Mathematical Analysis and Applications, 465(2), 762-794
Open this publication in new window or tab >>Geometric location of periodic points of 2-ramified power series
2018 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 465, no 2, p. 762-794Article in journal (Refereed) Published
Abstract [en]

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.

Elsevier, 2018
Geometry
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-64440 (URN)10.1016/j.jmaa.2018.05.009 (DOI)
Available from: 2017-05-29 Created: 2017-05-29 Last updated: 2020-01-13Bibliographically approved
Lindahl, K.-O. & Rivera-Letelier, J. (2016). Generic parabolic points are isolated in positive characteristic. Nonlinearity, 29(5), 1596-1621
Open this publication in new window or tab >>Generic parabolic points are isolated in positive characteristic
2016 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 29, no 5, p. 1596-1621Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2016
National Category
Other Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-46164 (URN)10.1088/0951-7715/29/5/1596 (DOI)000375008200006 ()2-s2.0-84963815028 (Scopus ID)
Funder
The Royal Swedish Academy of Sciences Available from: 2015-09-08 Created: 2015-09-08 Last updated: 2017-12-04Bibliographically approved
Lindahl, K.-O. & Rivera-Letelier, J. (2016). Optimal cycles in ultrametric dynamics and minimally ramified power series. Compositio Mathematica, 152(01), 187-222
Open this publication in new window or tab >>Optimal cycles in ultrametric dynamics and minimally ramified power series
2016 (English)In: Compositio Mathematica, ISSN 0010-437X, E-ISSN 1570-5846, Vol. 152, no 01, p. 187-222Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
London: London Mathematical Society, 2016
Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-31546 (URN)10.1112/S0010437X15007575 (DOI)000369432300006 ()2-s2.0-84955741084 (Scopus ID)
Note

Finansierad av SVeFUM Stiftelsen för svensk forskning och utbildning i Matematik, och MECESUP II, FONDECYT (Chile)

Available from: 2014-01-21 Created: 2014-01-21 Last updated: 2017-12-06Bibliographically approved
Lindahl, K.-O. (2013). The size of quadratic p-adic linearization disks. Advances in Mathematics, 248, 872-894
Open this publication in new window or tab >>The size of quadratic p-adic linearization disks
2013 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 248, p. 872-894Article in journal (Refereed) Published
Abstract [en]

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.

Elsevier, 2013
Keywords
Small divisors, Linearization, p-adic numbers, Ramification, Periodic points
National Category
Other Mathematics Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-27414 (URN)
Projects
Mathematical Modelling, Dynamical systems Available from: 2013-07-04 Created: 2013-07-04 Last updated: 2017-12-06Bibliographically approved
Lindahl, K.-O. (2010). Applied Algebraic Dynamics. P-Adic Numbers, Ultrametric Analysis, and Applications, 2(4), 360-362
Open this publication in new window or tab >>Applied Algebraic Dynamics
2010 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 4, p. 360-362Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Pleiades Publishing, Ltd., 2010
National Category
Mathematics Computational Mathematics Computer Sciences
Research subject
Natural Science, Mathematics; Mathematics, Applied Mathematics
Identifiers
urn:nbn:se:lnu:diva-10050 (URN)10.1134/S2070046610040084 (DOI)
Projects
Mathematical modeling: non-Archimedean Dynamics Available from: 2011-01-17 Created: 2011-01-17 Last updated: 2018-01-12Bibliographically approved
Lindahl, K.-O. (2010). Divergence and convergence of conjugacies in non-Archimedean dynamics. Contemporary Mathematics, 508, 89-109
Open this publication in new window or tab >>Divergence and convergence of conjugacies in non-Archimedean dynamics
2010 (English)In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 508, p. 89-109Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Providence, RI: Amer. Math. Soc., 2010
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-7589 (URN)
Projects
Mathematical modelling, Linnaeus University Available from: 2010-08-20 Created: 2010-08-20 Last updated: 2017-12-12Bibliographically approved
Lindahl, K.-O. & Zieve, M. (2010). On Hyperbolic Fixed Points in Ultrametric Dynamics. P-Adic Numbers, Ultrametric Analysis, and Applications, 2(3), 232-240
Open this publication in new window or tab >>On Hyperbolic Fixed Points in Ultrametric Dynamics
2010 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 2, no 3, p. 232-240Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Pleiades Publishing, Ltd, 2010
Keywords
dynamical system, linearization, conjugacy, ultrametric field
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-7587 (URN)10.1134/S2070046610030052 (DOI)
Projects
Mathematical modelling, Linnaeus University Available from: 2010-08-20 Created: 2010-08-20 Last updated: 2017-12-12Bibliographically approved
Lindahl, K.-O. (2009). Linearization in Ultrametric Dynamics in Fields of Characteristic Zero — Equal Characteristic Case. P-Adic Numbers, Ultrametric Analysis, and Applications, 1(4), 307-316
Open this publication in new window or tab >>Linearization in Ultrametric Dynamics in Fields of Characteristic Zero — Equal Characteristic Case
2009 (English)In: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 1, no 4, p. 307-316Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Berlin, Heidelberg, New York: Springer, 2009
Keywords
dynamical system, linearization, conjugacy, ultrametric field
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-7338 (URN)10.1134/S2070046609040049 (DOI)
Available from: 2010-02-25 Created: 2010-02-25 Last updated: 2017-12-12Bibliographically approved
Lindahl, K.-O. (2007). On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point. (Doctoral dissertation). Växjö: Växjö University Press
Open this publication in new window or tab >>On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point
2007 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Växjö: Växjö University Press, 2007. p. 160
Series
Acta Wexionensia, ISSN 1404-4307 ; 1404-4307
Keywords
dynamical system, linearization, conjugation, non-Archimedean field
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-1713 (URN)978-91-7636-574-8 (ISBN)
Public defence
2007-12-03, Homeros, Pelarhuset, Växjö, 13:15 (English)
Supervisors
Available from: 2007-11-28 Created: 2007-11-28 Last updated: 2014-02-20Bibliographically approved
Lindahl, K.-O. (2006). Uniqueness for p-adic meromorphic products [Review]. Mathematical Reviews
Open this publication in new window or tab >>Uniqueness for p-adic meromorphic products
2006 (English)In: Mathematical ReviewsArticle, book review (Other (popular science, discussion, etc.)) Published
Abstract [en]

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

Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:vxu:diva-4442 (URN)
Note
15 December 2006 Review för American Mathematical Society i serien Mathematical Reviews on the Web av artikeln Bekräftas med följande e-post X-Sieve: CMU Sieve 2.2 Date: Fri, 15 Dec 2006 00:17:41 -0500 From: mathrev@ams.org Subject: Review received (MR2232636) To: karl-olof.lindahl@vxu.se X-VXU-MailScanner-Information: Please contact the ISP for more information X-VXU-MailScanner: Found to be clean X-VXU-MailScanner-SpamCheck: not spam, SpamAssassin (not cached, score=-1.638, required 5, BAYES_00 -2.60, NO_REAL_NAME 0.96) X-VXU-MailScanner-From: webrev@ams.org http://www.ams.org/mathscinet/pdf/2232636.pdf?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&s4=boussaf&s5=&s6=&s7=&s8=All&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=3Available from: 2007-03-30 Created: 2007-03-30 Last updated: 2014-02-20Bibliographically approved
Identifiers
ORCID iD: orcid.org/0000-0002-7825-4428

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