Skip to main content
SpringerLink
Log in

The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We consider two numerical entropy-type invariants for actions of \({\mathbb{Z}^k}\) , invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. \({\mathbb{Z}^k,\,k \geq 2}\) actions on k + 1-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of \({{\mathbb Z}}\) actions the entropies of ergodic maximal rank actions take only countably many values. Our main result is the dichotomy that is best expressed under the assumption of weak mixing or, equivalently, no periodic factors: either both invariants vanish, or their values are bounded away from zero by universal constants. Furthermore, the lower bounds grow with dimension: for the first invariant (the Fried average entropy) exponentially, and for the second (the slow entropy) linearly.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Boyle M., Lind D.: Expansive subdynamics. Transactions of the American Mathematical Society, 349(1), 55–102 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Borevich Z.I., Shafarevich I.R.: Number Theory. Academic Press, New York (1966)

    Google Scholar 

  3. Fried D.: Entropy for smooth abelian actions. Proceedings of the American Mathematical Society, 87(1), 111–116 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. E. Friedman. Regulators and total positivity. In: Publicacions Matematiques. Proceedings of the Primeras Jornadas de Teoría de Números. (2007), pp. 119–130.

  5. G. Höhn, N.-P. Skoruppa. Un résultat de Schinzel. Journal de Théorie des Nombres de Bordeaux, 5 (1993), 185.

  6. Hu H.: Some ergodic properties of commuting diffeomorphisms. Ergodic Theory and Dynamical Systems 13, 73–100 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kalinin B., Katok A., Rodriguez Hertz F.: Nonuniform measure rigidity. Annals of Mathematics, 174, 361–400 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  8. Katok A.: Hyperbolic measures and commuting maps in low dimension. Discrete and Continuous Dynamical Systems, 2, 397–411 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Katok, S. Katok and K. Schmidt. Rigidity of measurable structure for \({{\mathbb Z}^d}\) actions by automorphisms of a torus. Commentarii Mathematici Helvetici, (4)77 (2002), 718–745.

  10. A. Katok and V. Nitica. Rigidity in Higher Rank Abelian Group Actions. I. Introduction and Cocycle Problem. Cambridge Tracts in Mathematics, Vol. 185. Cambridge University Press, Cambridge (2011).

  11. A. Katok and F. Rodriguez Hertz. Arithmeticity of maximal rank smooth abelian actions. http://www.math.psu.edu/katok_a/pub/arithmetic-draft-05-26.

  12. A. Katok and J.-P. Thouvenot. Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. Annales de l’Institut Henri Poincaré, (3)33 (1997), 323–338.

  13. Ledrappier F., Young L.-S.: The metric entropy of diffeomorphisms: part II: relations between entropy, exponents and dimension. Annals of Mathematics, 122(3), 540–574 (1985)

    Article  MathSciNet  Google Scholar 

  14. Meyer M., Pajor A.: Sections of the unit ball of \({I_p^n}\) . Journal of Functional Analysis, 80, 109–123 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ornstein D., Weiss B.: Entropy and isomorphism theorems for actions of amenable groups. Journal d’Analyse Mathématique, 48(1), 1–141 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pohst M., Zassenhaus H.: Algorithmic Algebraic Number Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  17. A. Schinzel. On the product of the conjugates outside the unit circle of an algebraic number. Acta Arithmetica 24 (1973), 385–399; Addendum: 26 (1974/1975), 329–331.

  18. I.Sh. Slavutskii. An estimator for the regulator of an algebraic field. (Russian. English summary) Mathematica Slovaca (3)41 (1991), 311–314.

  19. J. Voight. Enumeration of Totally Real Number Fields of Bounded Root Discriminant. Algorithmic Number Theory. Lecture Notes in Computer Science, Vol. 5011. Springer, Berlin (2008), pp. 268–281.

  20. Zimmert R.: Ideale kleiner Norm in Idealklassen and eine Regulatorabschätzung. Inventiones Mathematicae, 62, 367–380 (1981)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Anatole Katok, Svetlana Katok & Federico Rodriguez Hertz

Authors
  1. Anatole Katok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Svetlana Katok
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Federico Rodriguez Hertz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatole Katok.

Additional information

A. Katok: Based on research supported by NSF Grants DMS 1002554 and 1304830.

F. Rodriguez Hertz: Based on research supported by NSF Grant DMS 1201326.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katok, A., Katok, S. & Hertz, F.R. The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups. Geom. Funct. Anal. 24, 1204–1228 (2014). https://doi.org/10.1007/s00039-014-0284-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-014-0284-5

Keywords

Search

Navigation