Advertisement

Bowen’s Entropy for Endomorphisms of Totally Bounded Abelian Groups

  • Domingo Alcaraz
  • Dikran Dikranjan
  • Manuel SanchisEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 80)

Abstract

We say that the completion theorem holds for a uniform space \((X,\fancyscript{U})\) if, for every uniformly continuous function \(\alpha :(X,\fancyscript{U})\rightarrow (X,\fancyscript{U})\), the Bowen entropy of \(\alpha \) coincides with the Bowen entropy of \(\widetilde{\alpha }\), the extension of \(\alpha \) to the completion \((\widetilde{X},\widetilde{\fancyscript{U}})\) of \((X,\fancyscript{U})\). We study the completion theorem in the realm of abelian topological groups. Namely, we prove that it fails to be true in a drastic way by showing that every (abstract) abelian group \(G\) can be endowed with a totally bounded group topology \(\tau \) such that the topological group \((G,\tau )\) has endomorphisms of zero entropy whose extension to the Raĭkov completion of \((G,\tau )\) has infinite entropy. Our proof uses the structure theorems for abelian groups, the properties of the Bohr topology and Pontryagin duality. The case of metrizable groups is also analyzed in the case of Bernoulli shifts of finite groups, dense subgroups of the circle \(\mathbb T\) and the reals \(\mathbb R\). The key tool is the so-called \(e\)-supporting family of an endomorphism \(\alpha \) of an abelian metrizable group \(G\) with respect to a neighborhood \(U\) of the neutral element of \(G\). Several open questions are proposed.

Keywords

Bowen’s entropy Topological entropy Bohr compactification Totally bounded group Metrizable group 

Notes

Acknowledgments

The authors wish to thank the Editors for their kind invitation to participate in this Special Issue. Our contribution is dedicated to Prof. J. Kakol on occasion of his 60th birthday. The first authors was partially supported by “Progetti di Eccellenza 2011/12" of Fondazione CARIPARO. The third author was supported by the Spanish Ministry of Science and Education (Grant number MTM2011-23118), and by Bancaixa (Projecte P1-1B2011-30).

References

  1. 1.
    Adler, R.L., Konheim, A.G., McAndrew, M.H.: Topological entropy. Trans. Am. Math. Soc. 114, 309–319 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Arhangel’skii, A., Tkachenko, M.: Topological Groups and Related Structures. Atlantis Studies in Mathematics, vol. 1. Atlantis Press, Paris, World Scientific, Hackensack (2008)Google Scholar
  3. 3.
    Barbieri, G., Dikranjan, D., Milan, C., Weber, H.: Convergent sequences in precompact group topologies. Appl. Gen. Topology 6(2), 149–169 (2005)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Am. Math. Soc. 153, 401–414 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Comfort, W.W., van Mill, J.: Concerning connected, pseudocompact abelian groups. Topology Appl. 33(1), 21–45 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Comfort, W.W., Raczkowski, S.U., Trigos-Arrieta, F.J.: Making group topologies with, and without, convergent sequences. Appl. Gen. Topology 7(1), 109–124 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dikranjan, D., Giordano Bruno, A.: The connection between topological and algebraic entropy. Topology Appl. 159(13), 2980–2989 (2012)Google Scholar
  8. 8.
    Dikranjan, D., Goldsmith, B., Salce, L., Zanardo, P.: Algebraic entropy for Abelian groups. Trans. Am. Math. Soc. 361, 3401–3434 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hood, B.M.: Topological entropy and uniform spaces. J. Lond. Math. Soc. 8, 633–641 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Dikranjan, D., Prodanov, I.R., Stoyanov, L.N.: Topological groups (characters, dualities and minimal group topologies). Monographs and Textbooks in Pure and Applied Mathematics, vol. 130. Marcel Dekker Inc., New York (1990)Google Scholar
  11. 11.
    Dikranjan, D., Sanchis, M.: On the e-spectrum of a topological group (work in progress)Google Scholar
  12. 12.
    Dikranjan, D., Sanchis, M., Virili, S.: New and old facts about entropy in uniform spaces and topological groups. Topology Appl. 159, 1916–1942 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Dikranjan, D., Shakhmatov, D.: Algebraic structure of pseudocompact groups. Mem. Am. Math. Soc. 133(633), (1998)Google Scholar
  14. 14.
    Fuchs, L.: Infinite abelian groups, I. Pure and Applied Mathematics, vol. 36. Academic Press, New York (1970)Google Scholar
  15. 15.
    Fuchs, L.: Infinite abelian groups, II. Pure and Applied Mathematics, vol. 36-II. Academic Press, New York (1973)Google Scholar
  16. 16.
    Glicksberg, I.: Uniform boundedness for groups. Can. J. Math. 14, 269–276 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Goodwyn, L.W.: The product theorem for topological entropy. Trans. Am. Math. Soc. 158, 445–452 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Hewitt, E., Ross, K.A.: Abstract harmonic analysis, I: Structure of topological groups. Integration theory, group representations. Die Grundlehren der mathematischen Wissenschaften, Bd. 115, Academic Press, New York (1963)Google Scholar
  19. 19.
    Hofmann, K.K., Morris, S.A.: The structure of compact groups. A primer for the studentÃŚa handbook for the expert. De Gruyter Studies in Mathematics, vol. 25. De Gruyter, Berlin (2013)Google Scholar
  20. 20.
    Kimura, T.: Completion theorem for uniform entropy. Comm. Math. Univ. Carolin. 39(2), 389–399 (1998)zbMATHGoogle Scholar
  21. 21.
    Latora, V., Baranger, M., Rapisarda, A., Tsallis, C.: The rate of entropy increase at the edge of chaos. Phys. Lett. A 273(1–2), 97–103 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Pasynkov, B.A.: The coincidence of various definitions of dimensionality for locally bicompact groups. Dokl. Akad. Nauk SSSR 132, 1035–1037 (1960)MathSciNetGoogle Scholar
  23. 23.
    Pasynkov, B.A.: Almost-metrizable topological groups. Dokl. Akad. Nauk SSSR 161, 281–284 (1965). (in Russian)MathSciNetGoogle Scholar
  24. 24.
    Peters, J.: Entropy of discrete abelian groups. Adv. Math. 33(1), 1–13 (1979)CrossRefzbMATHGoogle Scholar
  25. 25.
    Peters, J.: Entropy of automorphisms on L.C.A. groups. Pac. J. Math. 96(2), 475–488 (1981)CrossRefzbMATHGoogle Scholar
  26. 26.
    Weil, A.: Sur les Spaces a Structure Uniforme and sur la Topologie Generale. Publ. Math. Univ. Strasbourg, Hermann (1937)Google Scholar
  27. 27.
    Weil, A.: L’intégration sur les groupes topologique et ses applications. Actualités Sci. Ind., vol. 869. Hermann, Paris (1950)Google Scholar
  28. 28.
    Weiss, M.D.: Algebraic and other entropies of group endomorphisms. Math. Syst. Theory 8(3), 243–248 (1974/75)Google Scholar
  29. 29.
    Yuzvinskiĭ, S.A.: Calculation of the entropy of a group-endomorphism. Sibirsk. Mat. Ž. 8, 230–239 (1967)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Domingo Alcaraz
    • 1
  • Dikran Dikranjan
    • 2
  • Manuel Sanchis
    • 3
    Email author
  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain
  2. 2.Dipartimento de Matematica e InformaticaUniversità di UdineUdineItaly
  3. 3.Institut de Matemàtiques i Aplicacions de Castelló (IMAC)Universitat Jaume I de CastellóCastellóSpain

Personalised recommendations