About the Algebraic Yuzvinski Formula

Anna Giordano Bruno; Simone Virili

Topological Algebra and its Applications (2015)

  • Volume: 3, Issue: 1, page 114-147
  • ISSN: 2299-3231

Abstract

top
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.

How to cite

top

Anna Giordano Bruno, and Simone Virili. "About the Algebraic Yuzvinski Formula." Topological Algebra and its Applications 3.1 (2015): 114-147. <http://eudml.org/doc/276015>.

@article{AnnaGiordanoBruno2015,
abstract = {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.},
author = {Anna Giordano Bruno, Simone Virili},
journal = {Topological Algebra and its Applications},
keywords = {algebraic entropy; topological entropy; endomorphism; Yuzvinski Formula; intrinsic entropy; Mahler measure; Yuzvinski formula; LCA group},
language = {eng},
number = {1},
pages = {114-147},
title = {About the Algebraic Yuzvinski Formula},
url = {http://eudml.org/doc/276015},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Anna Giordano Bruno
AU - Simone Virili
TI - About the Algebraic Yuzvinski Formula
JO - Topological Algebra and its Applications
PY - 2015
VL - 3
IS - 1
SP - 114
EP - 147
AB - 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.
LA - eng
KW - algebraic entropy; topological entropy; endomorphism; Yuzvinski Formula; intrinsic entropy; Mahler measure; Yuzvinski formula; LCA group
UR - http://eudml.org/doc/276015
ER -

References

top
  1. [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. Zbl0127.13102
  2. [2] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. Zbl0212.29201
  3. [3] D. Dikranjan and A. Giordano Bruno, Entropy on Abelian groups, submitted ; arXiv:1007.0533. Zbl06591707
  4. [4] D. Dikranjan and A. Giordano Bruno, The Bridge Theorem for totally disconnected LCA groups, Topology Appl. 169 (2014) no. 1, 21–32. [WoS] Zbl1322.37007
  5. [5] D. Dikranjan and A. Giordano Bruno, Limit free computation of entropy, Rend. Istit.Mat. Univ. Trieste 44 (2012), 297–312. Zbl1277.37031
  6. [6] D. Dikranjan and A. Giordano Bruno, Topological entropy and algebraic entropy for group endomorphisms, Proceedings ICTA2011 Islamabad, Pakistan July 4-10 2011 Cambridge Scientific Publishers, 133–214. Zbl1300.54002
  7. [7] D. Dikranjan and A. Giordano Bruno, The connection between topological and algebraic entropy, Topology Appl. 159 (2012) no.13, 2980–2989. [WoS] Zbl1256.54061
  8. [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] Zbl1247.37014
  9. [9] D. Dikranjan, A. Giordano Bruno, L. Salce and S.Virili, Intrinsic algebraic entropy, J. Pure Appl. Algebra 219 (2015) 2933– 2961. [WoS] Zbl06409606
  10. [10] D. Dikranjan, B. Goldsmith, L. Salce and P. Zanardo, Algebraic entropy for Abelian groups, Trans. Amer. Math. Soc. 361 (2009), 3401–3434. Zbl1176.20057
  11. [11] D. Dikranjan, K. Gong and P. Zanardo, Endomorphisms of Abelian groups with small algebraic entropy, Linear Algebra Appl. 439 (2013) no.7, 1894–1904. [WoS] Zbl1320.37013
  12. [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] Zbl1242.54005
  13. [13] M. Einsiedler and T. Ward, Ergodic Theory (with a view towards Number Theory), Graduate Texts in Mathematics Volume 259, 2011. [WoS] Zbl1206.37001
  14. [14] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer Verlag, 1999. Zbl0919.11064
  15. [15] A. Giordano Bruno and S. Virili, Algebraic Yuzvinski Formula, J. Algebra 423 (2015) 114–147. [WoS] Zbl06377569
  16. [16] E. Hewitt and K. A. Ross, Abstract harmonic analysis I, Springer-Verlag, Berlin-Heidelberg-New York, 1963. Zbl0115.10603
  17. [17] E. Hironaka, What is. . .Lehmer’s number?, Not. Amer. Math. Soc. 56 (2009) no. 3, 374–375. Zbl1163.11069
  18. [18] B. M. Hood, Topological entropy and uniform spaces, J. London Math. Soc. (2) 8 (1974), 633–641. [Crossref] Zbl0291.54051
  19. [19] I. Kaplansky, Infinite Abelian groups, University of Michigan Publications in Mathematics, no. 2, Ann Arbor, University of Michigan Press, 1954. Zbl0057.01901
  20. [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. Zbl0364.12015
  21. [21] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coeflcienten, Jour. Reine Angew. Math. 53 (1857), 173–175. 
  22. [22] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479. Zbl0007.19904
  23. [23] D. A. Lind and T. Ward, Automorphisms of solenoids and p-adic entropy, Ergod. Th. & Dynam. Sys. 8 (1988), 411–419. Zbl0634.22005
  24. [24] K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341–344. Zbl0105.06301
  25. [25] M. J. Mossinghoff, Lehmer’s Problem web page, http://www.cecm.sfu.ca/ mjm/Lehmer/lc.html. 
  26. [26] J. Peters, Entropy on discrete Abelian groups, Adv. Math. 33 (1979), 1–13. Zbl0421.28019
  27. [27] J. Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981) no.2, 475–488. Zbl0478.28010
  28. [28] F. Quadros Gouvêa, p-adic numbers: an introduction, Springer-Verlag, Berlin-Heidelberg-New York, 1997. 
  29. [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. Zbl1334.11081
  30. [30] L. N. Stojanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat. Ital. B 7 (1987) no. 3, 829–847. Zbl0648.22002
  31. [31] S. Virili, Entropy for endomorphisms of LCA groups, Topology Appl. 159 (2012) no.9, 2546–2556. Zbl1243.22007
  32. [32] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New-York, 1982. Zbl0475.28009
  33. [33] M. D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75) no.3, 243–248. Zbl0298.28014
  34. [34] A. Weil, Basic Number Theory, third edition, Springer Verlag, New York (1974). 
  35. [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. [36] S. A. Yuzvinski, Computing the entropy of a group endomorphism, Sibirsk.Mat. Z. 8 (1967), 230–239 (in Russian). English Translation: Siberian Math. J. 8 (1968), 172–178. 
  37. [37] P. Zanardo, Yuzvinski’s Formula, unpublished notes. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.