PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 3 | 1 |
Tytuł artykułu

About the Algebraic Yuzvinski Formula

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Algebraic Yuzvinski Formula expresses the algebraic entropy of an endomorphism of a finitedimensional rational vector space as the Mahler measure of its characteristic polynomial. In a recent paper, we have proved this formula, independently fromits counterpart – the Yuzvinski Formula – for the topological entropy proved by Yuzvinski in 1968. In this paper we first compare the proof of the Algebraic Yuzvinski Formula with a proof of the Yuzvinski Formula given by Lind and Ward in 1988, underlying the common ideas and the differences in the main steps. Then we describe several known applications of the Algebraic Yuzvinski Formula, and some related open problems are discussed. Finally,we give a new and purely algebraic proof of the Algebraic Yuzvinski Formula for the intrinsic algebraic entropy.
Wydawca
Rocznik
Tom
3
Numer
1
Opis fizyczny
Daty
otrzymano
2015-07-11
zaakceptowano
2015-08-30
online
2015-11-26
Twórcy
  • Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine,
    Italia
  • Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63,
    35121 Padova, Italia
Bibliografia
  • [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319.
  • [2] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.
  • [3] D. Dikranjan and A. Giordano Bruno, Entropy on Abelian groups, submitted ; arXiv:1007.0533.
  • [4] D. Dikranjan and A. Giordano Bruno, The Bridge Theorem for totally disconnected LCA groups, Topology Appl. 169 (2014)no. 1, 21–32.[WoS]
  • [5] D. Dikranjan and A. Giordano Bruno, Limit free computation of entropy, Rend. Istit.Mat. Univ. Trieste 44 (2012), 297–312.
  • [6] D. Dikranjan and A. Giordano Bruno, Topological entropy and algebraic entropy for group endomorphisms, ProceedingsICTA2011 Islamabad, Pakistan July 4-10 2011 Cambridge Scientific Publishers, 133–214.
  • [7] D. Dikranjan and A. Giordano Bruno, The connection between topological and algebraic entropy, Topology Appl. 159 (2012)no.13, 2980–2989.[WoS]
  • [8] D. Dikranjan and A. Giordano Bruno, The Pinsker subgroup of an algebraic flow, J. Pure Appl. Algebra 216 (2012) no.2,364–376.[WoS]
  • [9] D. Dikranjan, A. Giordano Bruno, L. Salce and S.Virili, Intrinsic algebraic entropy, J. Pure Appl. Algebra 219 (2015) 2933–2961.[WoS]
  • [10] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for Abelian groups, Trans. Amer. Math. Soc. 361(2009), 3401–3434.
  • [11] D. Dikranjan, K. Gong and P. Zanardo, Endomorphisms of Abelian groups with small algebraic entropy, Linear AlgebraAppl. 439 (2013) no.7, 1894–1904.[WoS]
  • [12] D. Dikranjan, M. Sanchis and S. Virili, New and old facts about entropy on uniform spaces and topological groups,Topology Appl. 159 (2012) no.7, 1916–1942.[WoS]
  • [13] M. Einsiedler and T. Ward, Ergodic Theory (with a view towards Number Theory), Graduate Texts in Mathematics Volume259, 2011.[WoS]
  • [14] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer Verlag, 1999.
  • [15] A. Giordano Bruno and S. Virili, Algebraic Yuzvinski Formula, J. Algebra 423 (2015) 114–147.[WoS]
  • [16] E. Hewitt and K. A. Ross, Abstract harmonic analysis I, Springer-Verlag, Berlin-Heidelberg-New York, 1963.
  • [17] E. Hironaka, What is. . .Lehmer’s number?, Not. Amer. Math. Soc. 56 (2009) no. 3, 374–375.
  • [18] B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc. (2) 8 (1974), 633–641.[Crossref]
  • [19] I. Kaplansky, Infinite Abelian groups, University of Michigan Publications in Mathematics, no. 2, Ann Arbor, University ofMichigan Press, 1954.
  • [20] N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, Second Edition, Graduate Texts in Mathematics 58,Springer-Verlag New York, Berlin, Heidelberg, Tokyo, 1984.
  • [21] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coeflcienten, Jour. Reine Angew. Math. 53 (1857), 173–175.
  • [22] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479.
  • [23] D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergod. Th. & Dynam. Sys. 8 (1988), 411–419.
  • [24] K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341–344.
  • [25] M. J. Mossinghoff, Lehmer’s Problem web page, http://www.cecm.sfu.ca/ mjm/Lehmer/lc.html.
  • [26] J. Peters, Entropy on discrete Abelian groups, Adv. Math. 33 (1979), 1–13.
  • [27] J. Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981) no.2, 475–488.
  • [28] F. Quadros Gouvêa, p-adic numbers: an introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1997.
  • [29] C. Smyth, TheMahler measure of algebraic numbers: a survey. Number theory and polynomials, 322–349, LondonMath.Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008.
  • [30] L. N. Stojanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat. Ital. B 7 (1987) no.3, 829–847.
  • [31] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl. 159 (2012) no.9, 2546–2556.
  • [32] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New-York, 1982.
  • [33] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75) no.3, 243–248.
  • [34] A. Weil, Basic Number Theory, third edition, Springer Verlag, New York (1974).
  • [35] S. A. Yuzvinski, Metric properties of endomorphisms of compact groups, Izv. Acad. Nauk SSSR, Ser.Mat. 29 (1965), 1295–1328 (in Russian). English Translation: Amer. Math. Soc. Transl. (2) 66 (1968), 63–98.
  • [36] S. A. Yuzvinski, Computing the entropy of a group endomorphism, Sibirsk.Mat. Z. 8 (1967), 230–239 (in Russian). EnglishTranslation: Siberian Math. J. 8 (1968), 172–178.
  • [37] P. Zanardo, Yuzvinski’s Formula, unpublished notes.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_taa-2015-0008
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.