On the essential hyperbolicity of sectional-Anosov flows

Bautista, S.; Morales, C.

doi:10.1090/proc/12686

On the essential hyperbolicity of sectional-Anosov flows
HTML articles powered by AMS MathViewer

by S. Bautista and C. A. Morales

Proc. Amer. Math. Soc. 144 (2016), 205-215

DOI: https://doi.org/10.1090/proc/12686

Published electronically: June 24, 2015

PDF | Request permission

Abstract:

We prove that every sectional-Anosov flow of a compact $3$-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications include a characterization of essential hyperbolicity, sensitivity to the initial conditions and a relationship between the topology of $M$ and the denseness of the basin of the singularities.

References

V. S. Afraĭmovič, V. V. Bykov, and L. P. Sil′nikov, The origin and structure of the Lorenz attractor, Dokl. Akad. Nauk SSSR 234 (1977), no. 2, 336–339 (Russian). MR 0462175
A. Araujo, Exisência de atratores hiperbólicos para difeomorfismos de superficies (in Portuguese), Prepublicações IMPA Série F, no. 23/88, 1988.
A. Arbieto, C. Morales, and L. Senos, On the sensitivity of sectional-Anosov flows, Math. Z. 270 (2012), no. 1-2, 545–557. MR 2875848, DOI 10.1007/s00209-010-0811-5
J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), no. 4, 332–334. MR 1157223, DOI 10.2307/2324899
S. Bautista and C. A. Morales, Lectures on sectional-Anosov flows. Preprint IMPA Série D 84 (2011).
S. Bautista and C. Morales, A sectional-Anosov connecting lemma, Ergodic Theory Dynam. Systems 30 (2010), no. 2, 339–359. MR 2599883, DOI 10.1017/S0143385709000157
S. Bautista and C. Morales, Characterizing omega-limit sets which are closed orbits, J. Differential Equations 245 (2008), no. 3, 637–652. MR 2422522, DOI 10.1016/j.jde.2007.11.007
S. Bautista and C. Morales, Existence of periodic orbits for singular-hyperbolic sets, Mosc. Math. J. 6 (2006), no. 2, 265–297, 406 (English, with English and Russian summaries). MR 2270614, DOI 10.17323/1609-4514-2006-6-2-265-297
S. Bautista, C. Morales, and M. J. Pacifico, On the intersection of homoclinic classes on singular-hyperbolic sets, Discrete Contin. Dyn. Syst. 19 (2007), no. 4, 761–775. MR 2342271, DOI 10.3934/dcds.2007.19.761
Christian Bonatti, António Pumariño, and Marcelo Viana, Lorenz attractors with arbitrary expanding dimension, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 8, 883–888 (English, with English and French summaries). MR 1485910, DOI 10.1016/S0764-4442(97)80131-0
S. Crovisier and E. R. Pujals, Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, Preprint arXiv:1011.3836v1 [math.DS] 16 Nov 2010.
John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133–160. MR 553966
John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
R. Labarca and M. J. Pacífico, Stability of singularity horseshoes, Topology 25 (1986), no. 3, 337–352. MR 842429, DOI 10.1016/0040-9383(86)90048-0
A. M. Lopez B., Finiteness of attractors and repellers on sectional hyperbolic sets, arXiv:1309.5558v1 [math.DS] 22 Sep 2013.
E. N. Lorenz, Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?, American Association for the Advancement of Science, December 29, 1972.
R. Metzger and C. Morales, Sectional-hyperbolic systems, Ergodic Theory Dynam. Systems 28 (2008), no. 5, 1587–1597. MR 2449545, DOI 10.1017/S0143385707000995
C. A. Morales, An improved sectional-Anosov closing lemma, Math. Z. 268 (2011), no. 1-2, 317–327. MR 2805437, DOI 10.1007/s00209-010-0673-x
C. A. Morales, Sectional-Anosov flows, Monatsh. Math. 159 (2010), no. 3, 253–260. MR 2600912, DOI 10.1007/s00605-008-0078-7
C. A. Morales, Singular-hyperbolic attractors with handlebody basins, J. Dyn. Control Syst. 13 (2007), no. 1, 15–24. MR 2288940, DOI 10.1007/s10883-006-9000-6
C. A. Morales, Examples of singular-hyperbolic attracting sets, Dyn. Syst. 22 (2007), no. 3, 339–349. MR 2354969, DOI 10.1080/14689360701210939
C. A. Morales, Strong stable manifolds for sectional-hyperbolic sets, Discrete Contin. Dyn. Syst. 17 (2007), no. 3, 553–560. MR 2276427, DOI 10.3934/dcds.2007.17.553
C. A. Morales and M. J. Pacifico, A dichotomy for three-dimensional vector fields, Ergodic Theory Dynam. Systems 23 (2003), no. 5, 1575–1600. MR 2018613, DOI 10.1017/S0143385702001621
C. A. Morales, M. J. Pacifico, and E. R. Pujals, Singular hyperbolic systems, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3393–3401. MR 1610761, DOI 10.1090/S0002-9939-99-04936-9
Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
Fabrizio Polo, Sensitive dependence on initial conditions and chaotic group actions, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2815–2826. MR 2644895, DOI 10.1090/S0002-9939-10-10286-X
David Ruelle, Microscopic fluctuations and turbulence, Phys. Lett. A 72 (1979), no. 2, 81–82. MR 589691, DOI 10.1016/0375-9601(79)90653-4
Y. G. Sinai, Chaos theory yesterday, today and tomorrow, J. Stat. Phys. 138 (2010), no. 1-3, 2–7. MR 2594887, DOI 10.1007/s10955-009-9912-0