Skip to main content
SpringerLink
Log in

Contributions to a Rigidity Conjecture

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

Let p and q be two relatively prime positive integers and μ a Borel probability measure invariant and ergodic by the semigroup generated by the action of both zp and zq. We analyse sufficient conditions to guarantee that μ is either the Lebesgue measure or supported on a periodic orbit. And extend the results for general expanding differentiable maps of the circle.

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. Berend, D.: Minimal sets on tori, Ergodic Theory Dynam. Systems 4(1984), 499-507.

    Google Scholar 

  2. Bowen, R. and Ruelle, D.: The ergodic theory of Axiom A flows, Invent. Math. 29(1975), 181-202.

    Google Scholar 

  3. Furstenberg, H.: Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Math. Sytems Theory 1(1967), 1-49.

    Google Scholar 

  4. Keane, M.: Strongly mixing g-measures, Invent. Math. 16(1972), 309-324.

    Google Scholar 

  5. Katok, A. and Spatzier, R. J.: Invariant measures for higher rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16(4) (1996), 751-778.

    Google Scholar 

  6. Lyons, R.: On measures simultaneously 2 and 3 invariant, Israel J. Math. 61(2) (1988), 219- 224.

    Google Scholar 

  7. Mañé, R.: The Hausdorff Dimension of Invariant Probabilities of Rational Maps, Lecture Notes in Math. 1331, Springer-Verlag, New York, 1988, pp. 86-117.

    Google Scholar 

  8. Mañé, R.: Ergodic Theory and Differentiable Dynamics, Springer-Verlag, New York, 1987.

    Google Scholar 

  9. Palis, J. and Yoccoz, J.-C.: Rigidity of centralizers of diffeomorphisms, Ann. Sci. Ecole Norm. Sup. 22(1989), 81-98.

    Google Scholar 

  10. Parry, W.: In general a degree two map is an automorphism, Contemp. Math. 135(1992), 335-338.

    Google Scholar 

  11. Rudolph: ×2 and ×3 invariant measures and entropy, Ergodic Theory Dynam. Systems 10(1990), 395-406.

    Google Scholar 

  12. Ruelle, D.: Statistical mechanics of a one dimensional lattice gas, Comm. Math. Phys. 9(1968), 267-278.

    Google Scholar 

  13. Shub, M.: Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91(1969), 175-199.

    Google Scholar 

  14. Young, L. S.: Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2(1982), 109-124.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Matemática, Faculdade de Ciências, Universidade do Porto, 4050, Porto, Portugal

    Maria Carvalho

Authors
  1. Maria Carvalho
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carvalho, M. Contributions to a Rigidity Conjecture. Acta Applicandae Mathematicae 53, 265–295 (1998). https://doi.org/10.1023/A:1005816204645

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005816204645

Search

Navigation