American Institute of Mathematical Sciences

September  2021, 41(9): 4477-4484. doi: 10.3934/dcds.2021044

Centralizers of partially hyperbolic diffeomorphisms in dimension 3

 1 Queen's University, Kingston, Ontario 2 Ohio State University, Columbus, Ohio

* Corresponding author: Andrey Gogolev

Received  January 2021 Revised  January 2021 Published  September 2021 Early access  March 2021

Fund Project: The first author was partially supported by the NSERC (Funding reference number RGPIN-2017-04592). The second author was partially supported by NSF DMS-1823150

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu [10] who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in [5,6].

Citation: Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4477-4484. doi: 10.3934/dcds.2021044
References:
 [1] T. Adachi, Closed orbits of an Anosov flow and the fundamental group, Proc. Amer. Math. Soc., 100 (1987), 595-598.  doi: 10.1090/S0002-9939-1987-0891171-5. [2] A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534. [3] A. Avila, M. Viana and A. Wilkinson, Absolute continuity, lyapunov exponents and rigidity Ⅱ: Systems with compact center leaves, to appear in Erg. Th. Dyn. Syst. (Katok memorial issue), 2019, available at http://www.math.uchicago.edu/ wilkinso/papers/AVW2.pdf [4] T. Barbot, De l'hyperbolique au globalement hyperbolique, Habilitation à diriger des recherches, Université Claude Bernard de Lyon, 2005, available at https://tel.archives-ouvertes.fr/tel-00011278. [5] T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part Ⅰ: The dynamically coherent case, arXiv e-prints (2019), arXiv:1908.06227. [6] T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part Ⅱ: Branching foliations, arXiv e-prints (2020), arXiv:2008.04871. [7] T. Barthelmé and A. Gogolev, A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows, MRL, 26 (2019). [8] M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen, 9 (1975), 9-19. [9] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489.  doi: 10.4007/annals.2010.171.451. [10] D. Damjanovic, A. Wilkinson and D. Xu, Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms, To appear in Duke Math. J., (2019), arXiv:1902.05201. [11] S. Fenley and R. Potrie, Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds, arXiv e-prints, 2018, arXiv:1809.02284. [12] A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443.  doi: 10.1017/etds.2016.50. [13] S. Hong and D. McCullough, Mapping class groups of 3-manifolds, then and now, Geometry and Topology Down Under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, 53–63. doi: 10.1090/conm/597/11768. [14] D. McCullough, Virtually geometrically finite mapping class groups of 3-manifolds, J. Differential Geom., 33 (1991), 1-65.  doi: 10.4310/jdg/1214446029. [15] J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754.  doi: 10.2307/2373755. [16] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque (1990), 268pp. [17] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.

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References:
 [1] T. Adachi, Closed orbits of an Anosov flow and the fundamental group, Proc. Amer. Math. Soc., 100 (1987), 595-598.  doi: 10.1090/S0002-9939-1987-0891171-5. [2] A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity Ⅰ: Geodesic flows, J. Eur. Math. Soc. (JEMS), 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534. [3] A. Avila, M. Viana and A. Wilkinson, Absolute continuity, lyapunov exponents and rigidity Ⅱ: Systems with compact center leaves, to appear in Erg. Th. Dyn. Syst. (Katok memorial issue), 2019, available at http://www.math.uchicago.edu/ wilkinso/papers/AVW2.pdf [4] T. Barbot, De l'hyperbolique au globalement hyperbolique, Habilitation à diriger des recherches, Université Claude Bernard de Lyon, 2005, available at https://tel.archives-ouvertes.fr/tel-00011278. [5] T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part Ⅰ: The dynamically coherent case, arXiv e-prints (2019), arXiv:1908.06227. [6] T. Barthelmé, S. R. Fenley, S. Frankel and R. Potrie, Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part Ⅱ: Branching foliations, arXiv e-prints (2020), arXiv:2008.04871. [7] T. Barthelmé and A. Gogolev, A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows, MRL, 26 (2019). [8] M. I. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen, 9 (1975), 9-19. [9] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489.  doi: 10.4007/annals.2010.171.451. [10] D. Damjanovic, A. Wilkinson and D. Xu, Pathology and asymmetry: Centralizer rigidity for partially hyperbolic diffeomorphisms, To appear in Duke Math. J., (2019), arXiv:1902.05201. [11] S. Fenley and R. Potrie, Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds, arXiv e-prints, 2018, arXiv:1809.02284. [12] A. Hammerlindl and R. Potrie, Partial hyperbolicity and classification: A survey, Ergodic Theory Dynam. Systems, 38 (2018), 401-443.  doi: 10.1017/etds.2016.50. [13] S. Hong and D. McCullough, Mapping class groups of 3-manifolds, then and now, Geometry and Topology Down Under, Contemp. Math., vol. 597, Amer. Math. Soc., Providence, RI, 2013, 53–63. doi: 10.1090/conm/597/11768. [14] D. McCullough, Virtually geometrically finite mapping class groups of 3-manifolds, J. Differential Geom., 33 (1991), 1-65.  doi: 10.4310/jdg/1214446029. [15] J. F. Plante, Anosov flows, Amer. J. Math., 94 (1972), 729-754.  doi: 10.2307/2373755. [16] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque (1990), 268pp. [17] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.
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