TY - JOUR
T1 - Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms
AU - Barthelmé, Thomas
AU - Fenley, Sergio R.
AU - Frankel, Steven
AU - Potrie, Rafael
N1 - Publisher Copyright:
© 2020 The Author(s). Published by Cambridge University Press.
PY - 2021/11/3
Y1 - 2021/11/3
N2 - We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen-Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.
AB - We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen-Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.
KW - 3-manifold topology
KW - classification
KW - foliations
KW - partial hyperbolicity
UR - https://www.scopus.com/pages/publications/85117687740
U2 - 10.1017/etds.2020.113
DO - 10.1017/etds.2020.113
M3 - Review article
AN - SCOPUS:85117687740
SN - 0143-3857
VL - 41
SP - 3227
EP - 3243
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
IS - 11
ER -