Logarithmic Bounds for Ergodic Sums of Certain Flows on the Torus: a Short Proof

• Jérôme Carrand ^[1]
  1. [1] Université de Paris
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 3, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We give a short proof that the ergodic sums of C1 observables for a C1 flow on T2 admitting a closed transversal curve whose Poincaré map has constant type rotation number have growth deviating at most logarithmically from a linear one. For this, we relate the latter integral to the Birkhoff sum of a well-chosen observable on the circle and use the Denjoy-Koksma inequality. We also give an example of a nonminimal flow satisfying the above assumptions

• Referencias bibliográficas
  • 1. Athanassopoulos, K.: Denjoy C1 diffeomorphisms of the circle and McDuff’s question. Expo. Math. 33(1), 48–66 (2015)
  • 2. Baladi, V.: There are no deviations for the ergodic averages of Giulietti–Liverani horocycle flows on the two-torus. Ergodic Theory and...
  • 3. Baladi, V., Tsujii, M.: Dynamical determinants and spectrum for hyperbolic diffeomorphisms. In:Geometric and probabilistic structures in...
  • 4. Bowen, R., Marcus, B.: Unique ergodicity for horocycle foliations. Israel J. Math. 26(1), 43–67 (1977)
  • 5. Carrand, J.: Explicit construction of non-linear pseudo-Anosov maps, with nonminimal invariant foliations. arXiv preprint arXiv:2104.11625,...
  • 6. Cornfeld, I.P., Fomin, S.V., Sinai, Y.G.: Ergodic Theory, vol. 245. Springer Science & Business Media, Berlin (2012)
  • 7. Coudène, Y.: Pictures of hyperbolic dynamical systems. Notices of the AMS 53(1), 8–13 (2006)
  • 8. Coudène, Y.: Ergodic Theory and Dynamical Systems. Springer, Berlin (2016)
  • 9. Dang, N.V., Rivière, G.: Pollicott-Ruelle spectrum and Witten Laplacians. J. Eur. Math. Soc. (JEMS) 23(6), 1797–1857 (2021)
  • 10. Dyatlov, S., Guillarmou, C.: Afterword: dynamical zeta functions for Axiom A flows. Bull. Amer. Math. Soc. (N.S.) 55(3), 337–342 (2018)
  • 11. Forni, G.: On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces. arXiv:2007.03144, (2020)
  • 12. Furstenberg, H.: The unique ergodicity of the horocycle flow. In Recent advances in topological dynamics (Proc. Conf., Yale Univ., New...
  • 13. Giulietti, P., Liverani, C.: Parabolic dynamics and Anisotropic Banach spaces. J. Eur. Math. Soc. 21(9), 2793–2858 (2019)
  • 14. Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. de l’IHÉS 49, 5–233 (1979)
  • 15. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems, vol. 54. Cambridge University Press, Cambridge (1997)
  • 16. Marcus, B.: Unique ergodicity of some flows related to Axiom A diffeomorphisms. Israel J. Math. 21(2–3), 111–132 (1975)
  • 17. Marcus, B.: Unique ergodicity of the horocycle flow: variable negative curvature case. Israel J. Math. 21(2–3), 133–144 (1975)
  • 18. Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)
  • 19. Yoccoz, J.-C.: Echanges d’intervalles. Cours Collège de France, https://www.college-de-france.fr/ media/jean-christophe-yoccoz/UPL8726_yoccoz05.pdf,...