Simultaneous Observability of Infinitely Many Strings and Beams
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Numeration 2016
Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Numeration 2016 Prague, Czech Republic, May 23–27 2016 Department of Mathematics FNSPE Czech Technical University in Prague Print: Česká technika — nakladatelství ČVUT Editor: Petr Ambrož [email protected] Katedra matematiky Fakulta jaderná a fyzikálně inženýrská České vysoké učení technické v Praze Trojanova 13 120 00 Praha 2 List of Participants Rafael Alcaraz Barrera [email protected] IME, University of São Paulo Karam Aloui [email protected] Institut Elie Cartan de Nancy / Faculté des Sciences de Sfax Petr Ambrož petr.ambroz@fjfi.cvut.cz FNSPE, Czech Technical University in Prague Hamdi Ammar [email protected] Sfax University Myriam Amri [email protected] Sfax University Hamdi Aouinti [email protected] Université de Tunis Simon Baker [email protected] University of Reading Christoph Bandt [email protected] University of Greifswald Attila Bérczes [email protected] University of Debrecen Anne Bertrand-Mathis [email protected] University of Poitiers Dávid Bóka [email protected] Eötvös Loránd University Horst Brunotte [email protected] Marta Brzicová [email protected] FNSPE, Czech Technical University in Prague Péter Burcsi [email protected] Eötvös Loránd University Francesco Dolce [email protected] Université Paris-Est Art¯urasDubickas [email protected] Vilnius University Lubomira Dvořáková [email protected] FNSPE, Czech -
On the Behavior of Mahler's Measure Under Iteration 1
ON THE BEHAVIOR OF MAHLER'S MEASURE UNDER ITERATION PAUL FILI, LUKAS POTTMEYER, AND MINGMING ZHANG Abstract. For an algebraic number α we denote by M(α) the Mahler measure of α. As M(α) is again an algebraic number (indeed, an algebraic integer), M(·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #OM (α), is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. 1. Introduction The Mahler measure of an algebraic number α with minimal polynomial f(x) = n anx + ··· + a0 2 Z[x] is defined as: n n Y Y M(α) = janj maxf1; jαijg = ±an αi: i=1 i=1 jαij>1 Qn where f(x) = an i=1(x − αi) 2 C[x]. It is clear that M(α) ≥ 1 is a real algebraic integer, and it follows from Kronecker's theorem that M(α) = 1 if and only if α is a root of unity. Moreover, we will freely use the facts that M(α) = M(β) whenever α and β have the same minimal polynomial, and that M(α) = M(α−1). D.H. Lehmer [10] famously asked in 1933 if the Mahler measure for an algebraic number which is not a root of unity can be arbitrarily close to 1. -
The Smallest Perron Numbers 1
MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers. -
New Families of Pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
AN ABSTRACT OF THE THESIS OF Hieu Trung Do for the degree of Doctor of Philosophy in Mathematics presented on May 27, 2016. Title: New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant. Abstract approved: Thomas A. Schmidt Translation surfaces can be viewed as polygons with parallel and equal sides identified. An affine homeomorphism φ from a translation surface to itself is called pseudo-Anosov when its derivative is a constant matrix in SL2(R) whose trace is larger than 2 in absolute value. In this setting, the eigendirections of this matrix defines the stable and unstable flow on the translation surface. Taking a transversal to the stable flows, the first return map of the flow induces an interval exchange transformation T . The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the lengths of the subintervals of T and their translations. This wedge product does not depend on the choice of transversal. We apply Veech's construction of pseudo-Anosov homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum H(2; 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in other strata. ©Copyright by Hieu Trung Do May 27, 2016 All Rights Reserved New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant by Hieu Trung Do A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Presented May 27, 2016 Commencement June 2016 Doctor of Philosophy thesis of Hieu Trung Do presented on May 27, 2016 APPROVED: Major Professor, representing Mathematics Chair of the Department of Mathematics Dean of the Graduate School I understand that my thesis will become part of the permanent collection of Oregon State University libraries. -
Combinatorics, Automata and Number Theory
Combinatorics, Automata and Number Theory CANT Edited by Val´erie Berth´e LIRMM - Univ. Montpelier II - CNRS UMR 5506 161 rue Ada, 34392 Montpellier Cedex 5, France Michel Rigo Universit´edeLi`ege, Institut de Math´ematiques Grande Traverse 12 (B 37), B-4000 Li`ege, Belgium 2 Number representation and finite automata Christiane Frougny Univ. Paris 8 and LIAFA, Univ. Paris 7 - CNRS UMR 7089 Case 7014, F-75205 Paris Cedex 13, France Jacques Sakarovitch LTCI, CNRS/ENST - UMR 5141 46, rue Barrault, F-75634 Paris Cedex 13, France. Contents 2.1 Introduction 5 2.2 Representation in integer base 8 2.2.1 Representation of integers 8 2.2.2 The evaluator and the converters 11 2.2.3 Representation of reals 18 2.2.4 Base changing 23 2.3 Representation in real base 23 2.3.1 Symbolic dynamical systems 24 2.3.2 Real base 27 2.3.3 U-systems 35 2.3.4 Base changing 43 2.4 Canonical numeration systems 48 2.4.1 Canonical numeration systems in algebraic number fields 48 2.4.2 Normalisation in canonical numeration systems 50 2.4.3 Bases for canonical numeration systems 52 2.4.4 Shift radix systems 53 2.5 Representation in rational base 55 2.5.1 Representation of integers 55 2.5.2 Representation of the reals 61 2.6 A primer on finite automata and transducers 66 2.6.1 Automata 66 2.6.2 Transducers 68 2.6.3 Synchronous transducers and relations 69 2.6.4 The left-right duality 71 3 4 Ch. -
Ito-Sadahiro Numbers Vs. Parry Numbers
Acta Polytechnica Vol. 51 No. 4/2011 Ito-Sadahiro numbers vs. Parry numbers Z. Mas´akov´a, E. Pelantov´a Abstract We consider a positional numeration system with a negative base, as introduced by Ito and Sadahiro. In particular, we focus on the algebraic properties of negative bases −β for which the corresponding dynamical system is sofic, which β happens, according to Ito and Sadahiro, if and only if the (−β)-expansion of − is eventually periodic. We call β +1 such numbers β Ito-Sadahiro numbers, and we compare their properties with those of Parry numbers, which occur in thesamecontextfortheR´enyi positive base numeration system. Keywords: numeration systems, negative base, Pisot number, Parry number. ∈AN 1 Introduction According to Parry, the string x1x2x3 ... rep- resents the β-expansion of a number x ∈ [0, 1) if and The expansion of a real number in the positional only if number system with base β>1, as defined by R´enyi [12] is closely related to the transformation ∗ xixi+1xi+2 ...≺ d (1) (2) T :[0, 1) → [0, 1), given by the prescription T (x):= β βx −#βx$.Everyx ∈ [0, 1) is a sum of the infinite for every i =1, 2, 3,... series D { | ∞ Condition (2) ensures that the set β = dβ(x) xi − ∈ } D x = , where x = #βT i 1(x)$ (1) x [0, 1) is shift invariant, and so the closure of β βi i N i=1 in A , denoted by Sβ, is a subshift of the full shift N for i =1, 2, 3,... A . The notion of β-expansion can naturally be ex- Directly from the definition of the transformation T tended to all non-negative real numbers: The expres- we can derive that the ‘digits’ x take values in the set i sion of a positive real number y in the form {0, 1, 2,...,%β&−1} for i =1, 2, 3,... -
On the Dichotomy of Perron Numbers and Beta-Conjugates Jean-Louis Verger-Gaugry
On the dichotomy of Perron numbers and beta-conjugates Jean-Louis Verger-Gaugry To cite this version: Jean-Louis Verger-Gaugry. On the dichotomy of Perron numbers and beta-conjugates. Monatshefte für Mathematik, Springer Verlag, 2008, 155, pp.277-299. 10.1007/s00605-008-0002-1. hal-00338280 HAL Id: hal-00338280 https://hal.archives-ouvertes.fr/hal-00338280 Submitted on 12 Nov 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the dichotomy of Perron numbers and beta-conjugates Jean-Louis Verger-Gaugry ∗ Abstract. Let β > 1 be an algebraic number. A general definition of a beta-conjugate i of β is proposed with respect to the analytical function fβ(z) = −1+ Pi≥1 tiz associated with the R´enyi β-expansion dβ(1) = 0.t1t2 . of unity. From Szeg¨o’s Theorem, we study the dichotomy problem for fβ(z), in particular for β a Perron number: whether it is a rational fraction or admits the unit circle as natural boundary. The first case of dichotomy meets Boyd’s works. We introduce the study of the geometry of the beta-conjugates with respect to that of the Galois conjugates by means of the Erd˝os-Tur´an approach and take examples of Pisot, Salem and Perron numbers which are Parry numbers to illustrate it. -
Special Classes of Algebraic Integers in Low-Dimensional Topology
Special classes of algebraic integers in low-dimensional topology Eriko Hironaka April 15, 2005 Abstract This note describes some open problems concerning distributions of special classes of real algebraic integers such as algebraic units, and Salem, P-V and Perron numbers. These special algebraic integers appear naturally as geometric invariants in low-dimensional topology. We relate properties of Salem, P-V and Perron to minimization problems in various geometric settings. 1 Introduction A complex number α ∈ C is an algebraic integer if it is a root of a monic integer polynomial. Two algebraic integers α and β are algebraically conjugate, written α ∼ β, if α and β satisfy the same irreducible monic integer polynomial. An algebraic integer α is an algebraic unit if α ∼ α−1. Let α be a real algebraic integer with α > 1. Consider all β ∼ α such that β 6= α: (i) if |β| < |α|, then α is a Perron number; (ii) if |β| < 1, then α is a P-V number; and (iii) if |β| ≤ 1 with at least one |β| = 1, then α is a Salem number. In this short note, we review definitions and known results concerning distributions of P-V, Salem and Perron numbers (Section 2), and relate them to geometric invariants in low-dimensional topology, including lengths of geodesics, growth rates of automatic groups, and homological and geometric dilatations of surface homeomorphisms (Section 3). 2 Distributions of algebraic integers and Lehmer’s problem Let P be the set of monic integer polynomials. Given f ∈ P, let Sf be the set of complex roots of + f counted with multiplicity, and let S (f) ⊂ Sf be the subset of points outside the unit circle C. -
A Proof of the Conjecture of Lehmer
A PROOF OF THE CONJECTURE OF LEHMER AND OF THE CONJECTURE OF SCHINZEL-ZASSENHAUS JEAN-LOUIS VERGER-GAUGRY September 13, 2017 Abstract. The conjecture of Lehmer is proved to be true. The proof mainly relies upon: (i) the properties of the Parry Upper functions f α (z) associated with the dynamical zeta functions ζ α (z) of the Rényi–Parry arithmetical dynamical systems, for α an algebraic integer α of house α greater than 1, (ii) the discovery of lenticuli of poles of ζ α (z) which uniformly equidistribute at the limit on a limit “lenticular" arc of the unit circle, when α tends to 1+, giving rise to a contin- uous lenticular minorant Mr( α ) of the Mahler measure M(α), (iii) the Poincaré asymptotic expansions of these poles and of this minorant Mr( α ) as a function of the dynamical degree. With the same arguments the conjecture of Schinzel- Zassenhaus is proved to be true. An inequality improving those of Dobrowolski and Voutier ones is obtained. The set of Salem numbers is shown to be bounded −1 from below by the Perron number θ31 =1.08545 ..., dominant root of the trinomial 1 z30 + z31. Whether Lehmer’s number is the smallest Salem number remains open.− − A lower bound for the Weil height of nonzero totally real algebraic num- bers, = 1, is obtained (Bogomolov property). For sequences of algebraic integers of Mahler6 ± measure smaller than the smallest Pisot number, whose houses have a dynamical degree tending to infinity, the Galois orbit measures of conjugates are proved to converge towards the Haar measure on z =1 (limit equidistribution). -
Uniform Distribution Theory 3 (2008), No.2, 157–190 Distribution Theory
uniform Uniform Distribution Theory 3 (2008), no.2, 157{190 distribution theory UNIFORM DISTRIBUTION OF GALOIS CONJUGATES AND BETA-CONJUGATES OF A PARRY NUMBER NEAR THE UNIT CIRCLE AND DICHOTOMY OF PERRON NUMBERS Jean-Louis Verger-Gaugry ABSTRACT. Concentration and equi-distribution, near the unit circle, in Solo- myak's set, of the union of the Galois conjugates and the beta-conjugates of a Parry number ¯ are characterized by means of the Erd}os-Tur¶anapproach, and its improvements by Mignotte and Amoroso, applied to the analytical function P i f¯ (z) = ¡1 + i¸1 tiz associated with the R¶enyi ¯-expansion d¯ (1) = 0:t1t2 ::: of unity. Mignotte's discrepancy function requires the knowledge of the factor- ization of the Parry polynomial of ¯. This one is investigated using theorems of Cassels, Dobrowolski, Pinner and Vaaler, Smyth, Schinzel in terms of cyclo- tomic, reciprocal non-cyclotomic and non-reciprocal factors. An upper bound of Mignotte's discrepancy function which arises from the beta-conjugates of ¯ which are roots of cyclotomic factors is linked to the Riemann hypothesis, following Amoroso. An equidistribution limit theorem, following Bilu's theorem, is formu- lated for the concentration phenomenon of conjugates of Parry numbers near the unit circle. Parry numbers are Perron numbers. Open problems on non-Parry Perron numbers are mentioned in the context of the existence of non-unique fac- torizations of elements of number ¯elds into irreducible Perron numbers (Lind). Communicated by Ilya Shkredov Contents 1. Introduction 158 2. Szeg}o'sTheorem and numeration 160 3. Parry polynomials, Galois- and beta-conjugates in Solomyak's set for a Parry number ¯ 161 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 11M99, 30B10, 12Y05. -
On the Behavior of Mahler's Measure Under Iteration
Monatshefte für Mathematik (2020) 193:61–86 https://doi.org/10.1007/s00605-020-01416-5 On the behavior of Mahler’s measure under iteration Paul A. Fili1 · Lukas Pottmeyer2 · Mingming Zhang1 Received: 21 November 2019 / Accepted: 9 April 2020 / Published online: 20 April 2020 © The Author(s) 2020 Abstract For an algebraic number α we denote by M(α) the Mahler measure of α.AsM(α) is again an algebraic number (indeed, an algebraic integer), M(·) is a self-map on Q, and therefore defines a dynamical system. The orbit size of α, denoted #OM (α),is the cardinality of the forward orbit of α under M. We prove that for every degree at least 3 and every non-unit norm, there exist algebraic numbers of every orbit size. We then prove that for algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also show that there exist algebraic units of larger degree with arbitrarily large but finite orbit size. Keywords Mahler measure · Lehmer’s problem · Algebraic numbers Mathematics Subject Classification 11R06 · 11R04 · 11R09 1 Introduction The Mahler measure of an algebraic number α with minimal polynomial f (x) = n an x +···+a0 ∈ Z[x], is defined as: Communicated by H. Bruin. B Lukas Pottmeyer [email protected] Paul A. Fili paul.fi[email protected] Mingming Zhang [email protected] 1 Oklahoma State University, Stillwater, OK 74078, USA 2 Universität Duisberg-Essen, 45117 Essen, Germany 123 62 P. A. Fili et al. n n M(α) =|an| max{1, |αi |} = ±an αi , i=1 i=1 |αi |>1 α ,...,α ( ) ( ) = n ( − α ) ∈ C[ ] where 1 n are the distinct roots of f x ; i.e. -
A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus Jean-Louis Verger-Gaugry
A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus Jean-Louis Verger-Gaugry To cite this version: Jean-Louis Verger-Gaugry. A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel- Zassenhaus. 2019. hal-02315014 HAL Id: hal-02315014 https://hal.archives-ouvertes.fr/hal-02315014 Preprint submitted on 14 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Manuscript File A Survey on the Conjecture of Lehmer and the Conjecture of Schinzel-Zassenhaus Jean-Louis Verger-Gaugry The Conjecture of Lehmer, and its refinement in 1965 by Schinzel, the Conjecture of Schinzel-Zassenhaus, amounts to a problem of universal minoration of the Mahler mea- sure, and of the height in higher dimension in Arithmetic Geometry. The objective of this Survey is to review the numerous minorations obtained in these two domains, in par- ticular Dobrowolski’s inequality, then to present the analogues of the problem of Lehmer in different contexts with various analogues of the Mahler measure and the height. The reformulation of the problem of Lehmer in other domains brings to light a certain number of situations generating integer polynomials for which the Problem of Lehmer is asked, and, if a nontrivial lower bound exists to the Mahler measure of these polynomials, the meaning and the realization of the situation of extremality.