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Normal forms for non-uniform contractions

BK: Supported in part by Simons Foundation grant 426243.
VS: Supported in part by NSF grant DMS-1301693.
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  • Let $f$ be a measure-preserving transformation of a Lebesgue space $(X,\mu)$ and let ${\mathscr{F}}$ be its extension to a bundle $\mathscr{E} = X \times {\mathbb{R}}^m$ by smooth fiber maps ${\mathscr{F}}_x : {\mathscr{E}}_x \to {\mathscr{E}}_{fx}$ so that the derivative of ${\mathscr{F}}$ at the zero section has negative Lyapunov exponents. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x$ on ${\mathscr{E}}_x$ for $\mu$-a.e. $x$ so that the maps ${\mathscr{P}}_x ={\mathscr{H}}_{fx} \circ {\mathscr{F}}_x \circ {\mathscr{H}}_x^{-1}$ are sub-resonance polynomials in a finite dimensional Lie group. Our construction shows that such ${\mathscr{H}}_x$ and ${\mathscr{P}}_x$ are unique up to a sub-resonance polynomial. As a consequence, we obtain the centralizer theorem that the coordinate change $\mathscr{H}$ also conjugates any commuting extension to a polynomial extension of the same type. We apply our results to a measure-preserving diffeomorphism $f$ with a non-uniformly contracting invariant foliation $W$. We construct a measurable system of smooth coordinate changes ${\mathscr{H}}_x: W_x \to T_xW$ such that the maps ${\mathscr{H}}_{fx} \circ f \circ {\mathscr{H}}_x^{-1}$ are polynomials of sub-resonance type. Moreover, we show that for almost every leaf the coordinate changes exist at each point on the leaf and give a coherent atlas with transition maps in a finite dimensional Lie group.

    Mathematics Subject Classification: Primary: 37D10, 37D25; Secondary: 37D30, 34C20.


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  • [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, SpringerVerlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
    [2] L. Arnold and X. Kedai, Normal forms for random diffeomorphisms, J. of Dynamics and Differential Equations, 4 (1992), 445-483.  doi: 10.1007/BF01053806.
    [3] L. Barreira and Ya. Pesin, Nonuniformly Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia of Mathematics and Its Applications, 115, Cambridge Univ. Press, Cambridge, 2007. doi: 10.1017/CBO9781107326026.
    [4] I. U. Bronstein and  A. Ya. KopanskiiSmooth Invariant Manifolds and Normal Forms, World Scientific, 1994.  doi: 10.1142/9789812798749.
    [5] Y. Fang, On the rigidity of quasiconformal Anosov flows, Ergodic Theory Dynam. Systems, 27 (2007), 1773-1802.  doi: 10.1017/S0143385707000326.
    [6] Y. FangP. Foulon and B. Hasselblatt, Zygmund strong foliations in higher dimension, J. of Modern Dynamics, 4 (2010), 549-569.  doi: 10.3934/jmd.2010.4.549.
    [7] R. Feres, A differential-geometric view of normal forms of contractions, in Modern Dynamical Systems and Applications, Cambridge University Press, 2004,103-121.
    [8] D. FisherB. Kalinin and R. Spatzier, Totally nonsymplectic Anosov actions on tori and nilmanifolds, Geometry and Topology, 15 (2011), 191-216.  doi: 10.2140/gt.2011.15.191.
    [9] A. GogolevB. Kalinin and V. Sadovskaya, Local rigidity for Anosov automorphisms(With appendix by R. de la Llave), Math. Research Letters, 18 (2011), 843-858.  doi: 10.4310/MRL.2011.v18.n5.a4.
    [10] M. Guysinsky, The theory of non-stationary normal forms, Ergodic Theory Dynam. Systems, 22 (2002), 845-862.  doi: 10.1017/S0143385702000421.
    [11] M. Guysinsky and A. Katok, Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations, Math. Research Letters, 5 (1998), 149-163.  doi: 10.4310/MRL.1998.v5.n2.a2.
    [12] B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, Proceedings of Symposia in Pure Mathematics, 69 (2001), 593-637.  doi: 10.1090/pspum/069/1858547.
    [13] B. Kalinin and A. Katok, Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori, J. of Modern Dynamics, 1 (2007), 123-146. 
    [14] B. KalininA. Katok and F. Rodriguez-Hertz, Nonuniform measure rigidity, Annals of Math., 174 (2011), 361-400.  doi: 10.4007/annals.2011.174.1.10.
    [15] A. Katok and J. Lewis, Local rigidity for certain groups of toral automorphisms, Israel J. Math., 75 (1991), 203-241.  doi: 10.1007/BF02776025.
    [16] A. Katok and F. Rodriguez-Hertz, Arithmeticity and topology of higher rank actions of Abelian groups, J. of Modern Dynamics, 10 (2016), 135-172.  doi: 10.3934/jmd.2016.10.135.
    [17] A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova, 216 (1997), Din. Sist. i Smezhnye Vopr., 292–319; translation in Proc. Steklov Inst. Math., 216 (1997), 287–314.
    [18] B. Kalinin and V. Sadovskaya, On local and global rigidity of quasiconformal Anosov diffeomorphisms, J. Inst. Math. Jussieu, 2 (2003), 567-582.  doi: 10.1017/S1474748003000161.
    [19] B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic Anosov Zk actions, Geometry and Topology, 10 (2006), 929-954.  doi: 10.2140/gt.2006.10.929.
    [20] B. Kalinin and V. Sadovskaya, Normal forms on contracting foliations: Smoothness and homogeneous structure, Geometriae Dedicata, 183 (2016), 181-194.  doi: 10.1007/s10711-016-0153-5.
    [21] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅰ. Characterization of measures satisfying Pesin's entropy formula, Annals of Math., 122 (1985), 509-539.  doi: 10.2307/1971328.
    [22] W. Li and K. Lu, Sternberg theorems for random dynamical systems, Communications on Pure and Applied Mathematics, 58 (2005), 941-988.  doi: 10.1002/cpa.20083.
    [23] K. Melnick, Nonstationary smooth geometric structures for contracting measurable cocycles, http://www.math.umd.edu/~kmelnick/.
    [24] D. Ruelle, Ergodic theory of differentiable dynamical systems, Publications Mathématiques de l'I.H.É.S., 50 (1979), 27-58. 
    [25] V. Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Research Letters, 12 (2005), 425-441.  doi: 10.4310/MRL.2005.v12.n3.a12.
    [26] S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. of Math., 79 (1957), 809-824.  doi: 10.2307/2372437.
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