A non-dynamically coherent example on T3
Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1023-1032.

In this paper we give the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle. The existence of such an example had been an open question since 1975 [2].

DOI : 10.1016/j.anihpc.2015.03.003
Mots-clés : Partial hyperbolicity, Dynamical coherence, Cohomological equations 
@article{AIHPC_2016__33_4_1023_0,
     author = {Rodriguez Hertz, F. and Rodriguez Hertz, M.A. and Ures, R.},
     title = {A non-dynamically coherent example on $ {\mathbb{T}}^{3}$},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1023--1032},
     publisher = {Elsevier},
     volume = {33},
     number = {4},
     year = {2016},
     doi = {10.1016/j.anihpc.2015.03.003},
     mrnumber = {3519530},
     zbl = {1380.37067},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2015.03.003/}
}
TY  - JOUR
AU  - Rodriguez Hertz, F.
AU  - Rodriguez Hertz, M.A.
AU  - Ures, R.
TI  - A non-dynamically coherent example on $ {\mathbb{T}}^{3}$
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2016
SP  - 1023
EP  - 1032
VL  - 33
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2015.03.003/
DO  - 10.1016/j.anihpc.2015.03.003
LA  - en
ID  - AIHPC_2016__33_4_1023_0
ER  - 
%0 Journal Article
%A Rodriguez Hertz, F.
%A Rodriguez Hertz, M.A.
%A Ures, R.
%T A non-dynamically coherent example on $ {\mathbb{T}}^{3}$
%J Annales de l'I.H.P. Analyse non linéaire
%D 2016
%P 1023-1032
%V 33
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2015.03.003/
%R 10.1016/j.anihpc.2015.03.003
%G en
%F AIHPC_2016__33_4_1023_0
Rodriguez Hertz, F.; Rodriguez Hertz, M.A.; Ures, R. A non-dynamically coherent example on $ {\mathbb{T}}^{3}$. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 4, pp. 1023-1032. doi : 10.1016/j.anihpc.2015.03.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.03.003/

[1] Brin, M.; Burago, D.; Ivanov, S. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus, J. Mod. Dyn., Volume 3 (2009), pp. 1–11 | MR | Zbl

[2] Brin, M.; Pesin, Y. Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 38 (1974), pp. 170–212 | MR | Zbl

[3] Burns, K.; Wilkinson, A. Dynamical coherence, accessibility and center bunching, Discrete Contin. Dyn. Syst., Volume 22 (2008), pp. 89–100 | MR | Zbl

[4] A. Hammerlindl, R. Potrie, Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group, preprint. | MR

[5] Hirsch, M.; Pugh, C.; Shub, M. Invariant Manifolds, Lect. Notes Math., vol. 583, Springer-Verlag, Berlin–New York, 1977 | DOI | MR | Zbl

[6] R. Potrie, Partial hyperbolicity and foliations in T3, preprint, 2012. | MR

[7] Rodriguez Hertz, F.; Rodriguez Hertz, M.; Ures, R. Tori with hyperbolic dynamics in 3-manifolds, J. Mod. Dyn., Volume 5 (2011), pp. 185–202 | MR | Zbl

[8] Rodriguez Hertz, F.; Rodriguez Hertz, M.; Ures, R. Partial hyperbolicity and ergodicity in dimension three, J. Mod. Dyn., Volume 2 (2008), pp. 187–208 | MR | Zbl

[9] F. Rodriguez Hertz, J. Rodriguez Hertz, R. Ures, Center-unstable foliations do not have compact leaves, preprint. | MR

[10] Shub, M. Topologically transitive diffeomorphisms on T4 , Lect. Notes Math., Volume 206 (1971), pp. 39–40 | DOI

[11] Smale, S. Differentiable dynamical systems, Bull. Am. Math. Soc., Volume 73 (1967), pp. 747–817 | DOI | MR | Zbl

[12] Wilkinson, A. Stable ergodicity of the time-one map of a geodesic flow, Ergod. Theory Dyn. Syst., Volume 18 (1998), pp. 1545–1587 | DOI | MR | Zbl

Cité par Sources :