Skip to main content Accessibility help

Volume hyperbolicity and wildness



It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$ -manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$ -manifold $M$ , the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$ -generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.



Hide All
[A] Abdenur, F.. Generic robustness of spectral decompositions. Ann. Sci. Éc. Norm. Supér. (4) 36(2) (2003), 213224.
[ABCDW] Abdenur, F., Bonatti, C., Crovisier, S., Díaz, L. J. and Wen, L.. Periodic points and homoclinic classes. Ergod. Th. & Dynam. Sys. 27(1) (2007), 122.
[B] Bonatti, C.. Towards a global view of dynamical systems, for the C 1 -topology. Ergod. Th. & Dynam. Sys. 31(4) (2011), 959993.
[BC] Bonatti, C. and Crovisier, S.. Récurrence et généricité. Invent. Math. 158(1) (2004), 33104.
[BC2] Bonatti, C. and Crovisier, S.. Center manifolds for partially hyperbolic sets without strong unstable connections. J. Inst. Math. Jussieu, published online 11 March 2015, doi:10.1017/S1474748015000055.
[BD1] Bonatti, C. and Díaz, L. J.. Persistent nonhyperbolic transitive diffeomorphisms. Ann. of Math. (2) 143(2) (1996), 357396.
[BD2] Bonatti, C. and Díaz, L. J.. Connexions hétéroclines et généricité d’une infinité de puits et de sources. Ann. Sci. Éc. Norm. Supér. (4) 32(1) (1999), 135150.
[BD3] Bonatti, C. and Díaz, L. J.. On maximal transitive sets of generic diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 96 (2003), 171197.
[BDK] Bonatti, C., Díaz, L. J. and Kiriki, S.. Stabilization of heterodimensional cycles. Nonlinearity 25(4) (2012), 931960.
[BDP] Bonatti, C., Díaz, L. J. and Pujals, E.. A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. (2) 158(2) (2003), 355418.
[BDV] Bonatti, C., Díaz, L. J. and Viana, M.. Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encyclopaedia of Mathematical Sciences (Mathematical Physics, III, 102) . Springer, Berlin, 2005.
[BLY] Bonatti, C., Li, M. and Yang, D.. On the existence of attractors. Trans. Amer. Math. Soc. 365 (2013), 13691391.
[BS] Bonatti, C. and Shinohara, K.. Flexible periodic points. Ergod. Th. & Dynam. Sys. 36(5) (2015), 13941422.
[BV] Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157193.
[Ca] Carvalho, M.. Sinai–Ruelle–Bowen measures for n-dimensional derived from Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 13(01) (1993), 2144.
[CP] Crovisier, S. and Pujals, E.. Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms. Invent. Math. 201(2) (2015), 385517.
[Gi] Gibbons, J. C.. One-dimensional basic sets in the three-sphere. Trans. Amer. Math. Soc. 164 (1972), 163178.
[Go] Gourmelon, N.. Adapted metrics for dominated splittings. Ergod. Th. & Dynam. Sys. 27(6) (2007), 18391849.
[HPS] Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, New York, 1977.
[M] Mañé, R.. Contributions to the stability conjecture. Topology 17(4) (1978), 383396.
[S] Shub, M.. Topologically transitive diffeomorphisms of T 4 . Proc. Sympos. Differential Equations and Dynamical Systems (Lecture Notes in Mathematics, 206) . Springer, Berlin, 1971, pp. 3940.
[W] Williams, R.. One-dimensional non-wandering sets. Topology 6 (1967), 473487.


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed