August 2017, 11: 263-312. doi: 10.3934/jmd.2017012

Exponential mixing and smooth classification of commuting expanding maps

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

2. 

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel

Received  August 07, 2016 Revised  February 04, 2017 Published  March 2017

Fund Project: RS: Supported in part by NSF grant DMS-1307164 and an Eisenbud Professorship at MSRI.
LY: Supported in part by a Postdoctoral Fellowship at MSRI, ERC grant AdG 267259, and ISF grant 0399180

We show that genuinely higher rank expanding actions of abelian semigroups on compact manifolds are $C.{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to Hölder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semigroup actions by nilmanifold endomorphisms, a result of independent interest. We then proceed similar to the case of higher rank Anosov actions.

Citation: Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
References:
[1]

E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511542879.

[2]

L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, in Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 57-263. doi: 10.1016/S1874-575X(06)80027-5.

[3]

J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, Thompson Book Co., 1967.

[4]

L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications. Part Ⅰ. Basic Theory and Examples, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, Cambridge, 1990.

[5]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topol. Methods Nonlinear Anal., 40 (2012), 111-136.

[6]

F. T. Farrell and L. E. Jones, Examples of expanding endomorphisms on exotic tori, Invent. Math., 45 (1978), 175-179. doi: 10.1007/BF01390271.

[7]

D. FisherB. Kalinin and R. Spatzier, Totally nonsymplectic Anosov actions on tori and nilmanifolds, Geom. Topol., 15 (2011), 191-216. doi: 10.2140/gt.2011.15.191.

[8]

D. FisherB. Kalinin and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds, with an appendix by J. F. Davis, J. Amer. Math. Soc., 26 (2013), 167-198. doi: 10.1090/S0894-0347-2012-00751-6.

[9]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes études Sci. Publ. Math., 53 (1981), 53-73.

[10]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphisms, Journal d'Analyse Mathématique, 123 (2014), 355-396. doi: 10.1007/s1185.

[11]

A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. doi: 10.1007/s11511-015-0130-0.

[12]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540. doi: 10.4007/annals.2012.175.2.2.

[13]

B. Green and T. Tao, On the quantitative distribution of polynomial nilsequences-erratum, Ann. of Math., 179 (2014), 1175-1183. doi: 10.4007/annals.2014.179.3.8.

[14]

B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001,593-637. doi: 10.1090/pspum/069.

[15]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory and Dynamical Systems, 16 (1996), 751-778. doi: 10.1017/S0143385700009081.

[16]

B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic anosov Zk actions, Geom. Topol., 10 (2006), 929-954. doi: 10.2140/gt.2006.10.929.

[17]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213. doi: 10.1017/S0143385704000215.

[18]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅰ. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539. doi: 10.2307/1971328.

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅱ. Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[20]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370. doi: 10.1090/S0002-9947-1977-0482849-4.

[21]

E. E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math., 56 (1952), 96-114. doi: 10.2307/1969769.

[22]

G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems, 21 (2001), 121-164. doi: 10.1017/S0143385701001109.

[23]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, American Journal of Mathematics, 91 (1969), 757-771. doi: 10.2307/2373350.

[24]

T. Radó, Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged, 2 (1925), 101-121.

[25]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, 68, Springer-Verlag, Berlin Heidelberg, 1972.

[26]

F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442. doi: 10.3934/jmd.2007.1.425.

[27]

F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math., 198 (2014), 165-209. doi: 10.1007/s00222-014-0499-y.

[28]

J. Rauch and M. Taylor, Regularity of functions smooth along foliations, and elliptic regularity, Journal of Functional Analysis, 225 (2005), 74-93. doi: 10.1016/j.jfa.2005.03.018.

[29]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 27-58.

[30]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. doi: 10.2307/2373276.

[31]

M. Shub, Expanding maps, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970,273-276.

[32]

R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics, J. Mod. Dyn., 10 (2016), 191-207. doi: 10.3934/jmd.2016.10.191.

[33]

A. N. Starkov, The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus, J. Math. Sci. (New York), 95 (1999), 2576-2582. doi: 10.1007/BF02169057.

[34]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dynam. Systems, 6 (1986), 449-473. doi: 10.1017/S0143385700003606.

[35]

P. Walters, Conjugacy properties of affine transformations of nilmanifolds, Math. Systems Theory, 4 (1970), 327-333. doi: 10.1007/BF01704076.

[36]

R. F. Williams, Expanding attractors, Publications Mathématiques de l'IHéS, 43 (1974), 169-203.

show all references

References:
[1]

E. Bombieri and W. Gubler, Heights in Diophantine Geometry, New Mathematical Monographs, 4, Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511542879.

[2]

L. Barreira and Y. Pesin, Smooth ergodic theory and nonuniformly hyperbolic dynamics, in Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 57-263. doi: 10.1016/S1874-575X(06)80027-5.

[3]

J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, Thompson Book Co., 1967.

[4]

L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications. Part Ⅰ. Basic Theory and Examples, Cambridge Studies in Advanced Mathematics, 18, Cambridge University Press, Cambridge, 1990.

[5]

K. Dekimpe, What an infra-nilmanifold endomorphism really should be..., Topol. Methods Nonlinear Anal., 40 (2012), 111-136.

[6]

F. T. Farrell and L. E. Jones, Examples of expanding endomorphisms on exotic tori, Invent. Math., 45 (1978), 175-179. doi: 10.1007/BF01390271.

[7]

D. FisherB. Kalinin and R. Spatzier, Totally nonsymplectic Anosov actions on tori and nilmanifolds, Geom. Topol., 15 (2011), 191-216. doi: 10.2140/gt.2011.15.191.

[8]

D. FisherB. Kalinin and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds, with an appendix by J. F. Davis, J. Amer. Math. Soc., 26 (2013), 167-198. doi: 10.1090/S0894-0347-2012-00751-6.

[9]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes études Sci. Publ. Math., 53 (1981), 53-73.

[10]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphisms, Journal d'Analyse Mathématique, 123 (2014), 355-396. doi: 10.1007/s1185.

[11]

A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms, Acta Math., 215 (2015), 127-159. doi: 10.1007/s11511-015-0130-0.

[12]

B. Green and T. Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math., 175 (2012), 465-540. doi: 10.4007/annals.2012.175.2.2.

[13]

B. Green and T. Tao, On the quantitative distribution of polynomial nilsequences-erratum, Ann. of Math., 179 (2014), 1175-1183. doi: 10.4007/annals.2014.179.3.8.

[14]

B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001,593-637. doi: 10.1090/pspum/069.

[15]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory and Dynamical Systems, 16 (1996), 751-778. doi: 10.1017/S0143385700009081.

[16]

B. Kalinin and V. Sadovskaya, Global rigidity for totally nonsymplectic anosov Zk actions, Geom. Topol., 10 (2006), 929-954. doi: 10.2140/gt.2006.10.929.

[17]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213. doi: 10.1017/S0143385704000215.

[18]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅰ. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math., 122 (1985), 509-539. doi: 10.2307/1971328.

[19]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅱ. Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574. doi: 10.2307/1971329.

[20]

R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., 229 (1977), 351-370. doi: 10.1090/S0002-9947-1977-0482849-4.

[21]

E. E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math., 56 (1952), 96-114. doi: 10.2307/1969769.

[22]

G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems, 21 (2001), 121-164. doi: 10.1017/S0143385701001109.

[23]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, American Journal of Mathematics, 91 (1969), 757-771. doi: 10.2307/2373350.

[24]

T. Radó, Über den Begriff der Riemannschen Fläche, Acta Litt. Sci. Szeged, 2 (1925), 101-121.

[25]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2. Folge, 68, Springer-Verlag, Berlin Heidelberg, 1972.

[26]

F. Rodriguez Hertz, Global rigidity of certain abelian actions by toral automorphisms, J. Mod. Dyn., 1 (2007), 425-442. doi: 10.3934/jmd.2007.1.425.

[27]

F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank abelian Anosov algebraic actions, Invent. Math., 198 (2014), 165-209. doi: 10.1007/s00222-014-0499-y.

[28]

J. Rauch and M. Taylor, Regularity of functions smooth along foliations, and elliptic regularity, Journal of Functional Analysis, 225 (2005), 74-93. doi: 10.1016/j.jfa.2005.03.018.

[29]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 27-58.

[30]

M. Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math., 91 (1969), 175-199. doi: 10.2307/2373276.

[31]

M. Shub, Expanding maps, in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R. I., 1970,273-276.

[32]

R. Spatzier, On the work of Rodriguez Hertz on rigidity in dynamics, J. Mod. Dyn., 10 (2016), 191-207. doi: 10.3934/jmd.2016.10.191.

[33]

A. N. Starkov, The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus, J. Math. Sci. (New York), 95 (1999), 2576-2582. doi: 10.1007/BF02169057.

[34]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dynam. Systems, 6 (1986), 449-473. doi: 10.1017/S0143385700003606.

[35]

P. Walters, Conjugacy properties of affine transformations of nilmanifolds, Math. Systems Theory, 4 (1970), 327-333. doi: 10.1007/BF01704076.

[36]

R. F. Williams, Expanding attractors, Publications Mathématiques de l'IHéS, 43 (1974), 169-203.

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