Abstract
In this paper we consider a diffeomorphism f of a compact manifold \(\mathcal {M}\) which contracts an invariant foliation W with smooth leaves. If the differential of f on TW has narrow band spectrum, there exist coordinates \({\mathcal {H}}_x:W_x\rightarrow T_xW\) in which \(f|_W\) has polynomial form. We present a modified approach that allows us to construct maps \({\mathcal {H}}_x\) that depend smoothly on x along the leaves of W. Moreover, we show that on each leaf they give a coherent atlas with transition maps in a finite dimensional Lie group. Our results apply, in particular, to \(C^1\)-small perturbations of algebraic systems.
Similar content being viewed by others
References
Bronstein, I.U., Kopanskii, AYa.: Smooth Invariant Manifolds and Normal Forms. World Scientific, Singapore (1994)
Fang, Y.: On the rigidity of quasiconformal Anosov flows. Ergod. Theory Dyn. Syst. 27(6), 1773–1802 (2007)
Fang, Y., Foulon, P., Hasselblatt, B.: Zygmund strong foliations in higher dimension. J. Mod. Dyn. 4(3), 549–569 (2010)
Feres, R.: The invariant connection of 1/2-pinched Anosov diffeomorphism and rigidity. Pac. J. Math. 171(1), 139–155 (1995)
Feres, R.: A differential-geometric view of normal forms of contractions. In: Brin, M., Hasselblatt, B., Pesin, Y. (eds.) Modern Dynamical Systems and Applications, pp. 103–121. Cambridge University Press, Cambridge (2004)
Fisher, D., Kalinin, B., Spatzier, R.: Totally non-symplectic Anosov actions on tori and nilmanifolds. Geom. Topol. 15, 191–216 (2011)
Gogolev, A., Kalinin, B., Sadovskaya, V.: Local rigidity for Anosov automorphisms. (with Appendix by R. de la Llave). Math. Res. Lett. 18(05), 843–858 (2011)
Guysinsky, M.: The theory of non-stationary normal forms. Ergod. Theory Dyn. Syst. 22(3), 845–862 (2002)
Guysinsky, M., Katok, A.: Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. Math. Res. Lett. 5, 149–163 (1998)
Hirsch, M., Pugh, C., Shub, M.: Invariant Manifolds. Springer, New York (1977)
Kalinin, B., Katok, A., Rodriguez-Hertz, F.: Nonuniform measure rigidity. Ann. Math. 174(1), 361–400 (2011)
Katok, A., Lewis, J.: Local rigidity for certain groups of toral automorphisms. Isr. J. Math. 75, 203–241 (1991)
Katok, A., Spatzier, R.: Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Tr. Mat. Inst. Steklova 216, Din. Sist. i Smezhnye Vopr., 292319; translation in Proc. Steklov Inst. Math. 1997, no. 1 (216), 287–314 (1997)
Kalinin, B., Sadovskaya, V.: On local and global rigidity of quasiconformal Anosov diffeomorphisms. J. Inst. Math. Jussieu 2(4), 567–582 (2003)
Kalinin, B., Sadovskaya, V.: Global rigidity for totally nonsymplectic Anosov \({\mathbb{Z}}^k\) actions. Geom. Topol. 10, 929–954 (2006)
Kalinin, B., Sadovskaya, V.: On classification of resonance-free Anosov \({\mathbb{Z}}^k\) actions. Mich. Math. J. 55(3), 651–670 (2007)
Pesin, Ya.: Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lectures in Advanced Mathematics, EMS (2004)
Pugh, C., Shub, M., Wilkinson, A.: Hölder foliations. Duke Math. J. 86(3), 517–546 (1997)
Sadovskaya, V.: On uniformly quasiconformal Anosov systems. Math. Res. Lett. 12(3), 425–441 (2005)
Sternberg, S.: Local contractions and a theorem of Poincaré. Am. J. Math. 79, 809–824 (1957)
Author information
Authors and Affiliations
Corresponding author
Additional information
Victoria Sadovskaya: Supported in part by NSF Grant DMS-1301693.
Rights and permissions
About this article
Cite this article
Kalinin, B., Sadovskaya, V. Normal forms on contracting foliations: smoothness and homogeneous structure. Geom Dedicata 183, 181–194 (2016). https://doi.org/10.1007/s10711-016-0153-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-016-0153-5