August 2017, 11: 341-368. doi: 10.3934/jmd.2017014

Normal forms for non-uniform contractions

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

Received  May 12, 2016 Revised  March 03, 2017 Published  April 2017

Fund Project: BK: Supported in part by Simons Foundation grant 426243.
VS: Supported in part by NSF grant DMS-1301693

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.

Citation: Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014
References:
[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. Kopanskii, Smooth 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.

show all references

References:
[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. Kopanskii, Smooth 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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