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Partially hyperbolic surface endomorphisms

Part of: Dynamical systems with hyperbolic behavior Differential topology Smooth dynamical systems: general theory

Published online by Cambridge University Press:  03 September 2019

LAYNE HALL  and
ANDY HAMMERLINDL
LAYNE HALL
Affiliation:
School of Mathematical Sciences, Monash University, Victoria3800, Australia email layne.hall@monash.edu, andy.hammerlindl@monash.edu
ANDY HAMMERLINDL
Affiliation:
School of Mathematical Sciences, Monash University, Victoria3800, Australia email layne.hall@monash.edu, andy.hammerlindl@monash.edu
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Abstract

We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^{2}$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^{2}$ which does not admit a centre foliation.

Keywords

partial hyperbolicitynon-invertible dynamicsdynamical coherence

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C05: Smooth mappings and diffeomorphisms 57R30: Foliations; geometric theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 1 , January 2021 , pp. 272 - 282
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Copyright
© Cambridge University Press, 2019

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References

Brin, M., Burago, D. and Ivanov, S.. On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group. Modern Dynamical Systems and Applications. Eds. Hasselblatt, B., Brin, M. and Pesin, Y.. Cambridge University Press, New York, 2004, pp. 307–312.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475–508.10.1016/j.top.2004.10.009Google Scholar
Crovisier, S. and Potrie, R.. Introduction to partially hyperbolic dynamics. Unpublished course notes, School on Dynamical Systems, ICTP, Trieste, 2015. available online.Google Scholar
Franks, J.. Anosov diffeomorphisms. Proc. Sympos. Pure Math. 14 (1970), 61–93.10.1090/pspum/014/0271990Google Scholar
Fuller, F. B.. On the surface of section and periodic trajectories. Amer. J. Math. 87 (1965), 473–480.10.2307/2373014Google Scholar
Hammerlindl, A.. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete Contin. Dyn. Syst. 33(8) (2013), 3641–3669.10.3934/dcds.2013.33.3641Google Scholar
Hammerlindl, A.. Ergodic components of partially hyperbolic systems. Comment. Math. Helv. 92(1) (2017), 131–184.10.4171/CMH/409Google Scholar
Hammerlindl, A. and Potrie, R.. Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group. J. Topol. 8(3) (2015), 842–870.Google Scholar
Hammerlindl, A. and Potrie, R.. Partial hyperbolicity and classification: a survey. Ergod. Th. & Dynam. Sys. 38(2) (2016), 1–43.Google Scholar
He, B., Shi, Y. and Wang, X.. Dynamical coherence of specially absolutely partially hyperbolic endomorphisms on 𝕋2. Nonlinearity 32(5) (2019), 1695–1704.10.1088/1361-6544/ab004dGoogle Scholar
Mañé, R. and Pugh, C.. Stability of endomorphisms. Dynamical Systems (Warwick, 1974). Springer, Berlin, 1975, pp. 175–184.Google Scholar
Potrie, R.. Partial hyperbolicity and foliations in 𝕋3. J. Mod. Dyn. 9 (2015), 81–121.Google Scholar
Przytycki, F.. Anosov endomorphisms. Studia Math. 3(58) (1976), 249–285.10.4064/sm-58-3-249-285Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. A non-dynamically coherent example on 𝕋3. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(4) (2016), 1023–1032.Google Scholar