Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T04:40:37.776Z Has data issue: false hasContentIssue false
  • English
  • Français

On the non-robustness of intermingled basins

Published online by Cambridge University Press:  04 July 2016

RAÚL URES  and
CARLOS H. VÁSQUEZ
RAÚL URES
Affiliation:
IMERL, Facultad de Ingeniería, Universidad de la República, CC 30, Montevideo, Uruguay email [email protected]
CARLOS H. VÁSQUEZ
Affiliation:
Instituto de Matemática, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that it is possible to construct a partially hyperbolic diffeomorphism on the 3-torus in a similar way to Kan’s example. It has two hyperbolic physical measures with intermingled basins supported on two embedded tori with Anosov dynamics. A natural question is how robust is the intermingled basin phenomenon for diffeomorphisms defined on boundaryless manifolds? In this work we study partially hyperbolic diffeomorphisms on the 3-torus and show that the intermingled basin phenomenon is not robust.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 384 - 400
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andersson, M. and Vásquez, C. H.. On mostly expanding diffeomorphisms. Preprint, December 2015, arXiv:1512.01046.Google Scholar
Barreira, L. and Pesin, Y.. Lyapunov Exponents and Smooth Ergodic Theory (University Lecture Series, 23) . American Mathematical Society, Providence, RI, 2002.Google Scholar
Bonatti, C., Díaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III) . Springer, Berlin, 2005.Google Scholar
Bonifant, A. and Milnor, J.. Schwarzian derivatives and cylinder maps. Holomorphic Dynamics and Renormalization (Fields Institute Communications, 53) . American Mathematical Society, Providence, RI, 2008, pp. 1–21.Google Scholar
Brin, M., Burago, D. and Ivanov, S.. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3(1) (2009), 1–11.Google Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.Google Scholar
Burns, K., Dolgopyat, D. and Pesin, Y.. Partial hyperbolicity, Lyapunov exponents and stable ergodicity. J. Stat. Phys. 108(5–6) (2002), 927–942, dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.CrossRefGoogle Scholar
Burns, K., Rodriguez Hertz, F., Rodriguez Hertz, J., Talitskaya, A. and Ures, R.. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 75–88.Google Scholar
Fathi, A., Herman, M. and Yoccoz, J.. A proof of Pesin’s stable manifold theorem. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007) . Springer, Berlin, 1983, pp. 177–215.CrossRefGoogle Scholar
Fayad, B.. Topologically mixing flows with pure point spectrum. Dynamical Systems. Part II (Pubblicazioni del Centro di Ricerca Matematica Ennio Giorgi) . Scuola Normale Superiore, Pisa, 2003, pp. 113–136.Google Scholar
Hammerlindl, A.. Leaf conjugacies on the torus. PhD Thesis, University of Toronto, Canada, 2009, ProQuest LLC, Ann Arbor, MI.Google Scholar
Hammerlindl, A.. Leaf conjugacies on the torus. Ergod. Th. & Dynam. Sys. 33(3) (2013), 896–933.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in three-dimensional nilmanifolds. J. Lond. Math. Soc. (2) 89(3) (2014), 853–875.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Ilyashenko, Y., Kleptsyn, V. and Saltykov, P.. Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins. J. Fixed Point Theory Appl. 3(2) (2008), 449–463.Google Scholar
Kan, I.. Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin. Bull. Amer. Math. Soc. (N.S.) 31(1) (1994), 68–74.Google Scholar
Kleptsyn, V. and Saltykov, P.. On C 2 -stable effects of intermingled basins of attractors in classes of boundary-preserving maps. Trans. Moscow Math. Soc. 72 (2011), 193–217.CrossRefGoogle Scholar
Mañé, R.. Ergodic Theory and Differentiable Dynamics (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8) . Springer, Berlin, 1987.CrossRefGoogle Scholar
Melbourne, I. and Windsor, A.. A C ∞ diffeomorphism with infinitely many intermingled basins. Ergod. Th. & Dynam. Sys. 25(6) (2005), 1951–1959.CrossRefGoogle Scholar
Niţică, V. and Török, A.. An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one. Topology 40(2) (2001), 259–278.CrossRefGoogle Scholar
Okunev, A.. Milnor attractors of circle skew products. Preprint, August 2015, arXiv:1508.02132.Google Scholar
Oseledec, V.. A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems. Tr. Mosk. Mat. Obs. 19 (1968), 179–210.Google Scholar
Pesin, Y.. Families of invariant manifolds that correspond to nonzero characteristic exponents. Izv. Akad. Nauk SSSR Ser. Mat. 40(6) (1976), 1332–1379, 1440.Google Scholar
Pesin, Y.. Characteristic Ljapunov exponents, and smooth ergodic theory. Uspekhi Mat. Nauk 32(4(196)) (1977), 55–112, 287.Google Scholar
Pesin, Y. and Sinai, Y.. Gibbs measures for partially hyperbolic attractors. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 417–438.Google Scholar
Potrie, R.. A few remarks on partially hyperbolic diffeomorphisms of T3 isotopic to Anosov. J. Dynam. Differential Equations 26(3) (2014), 805–815.CrossRefGoogle Scholar
Potrie, R.. Partial hyperbolicity and foliations in T3 . J. Mod. Dyn. 9 (2015), 81–121.CrossRefGoogle Scholar
Pugh, C. and Shub, M.. Ergodic attractors. Trans. Amer. Math. Soc. 312(1) (1989), 1–54.CrossRefGoogle Scholar
Rodriguez Hertz, F., Rodriguez Hertz, J., Tahzibi, A. and Ures, R.. Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys. 306(1) (2011), 35–49.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, J. and Ures, R.. A non-dynamically coherent example on $\mathbb{T}^{3}$ . Ann. Inst. H. Poincaré Anal. Non Linéaire to appear.Google Scholar
Ures, R.. Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part. Proc. Amer. Math. Soc. 140(6) (2012), 1973–1985.Google Scholar
Dolgopyat, D., Viana, M. and Yang, J.. Geometric and measure-theoretical structures of maps with mostly contracting center. Comm. Math. Phys. 341(3) (2016), 991–1014.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.Google Scholar
Young, L.. What are SRB measures, and which dynamical systems have them? J. Stat. Phys. 108(5–6) (2002), 733–754.Google Scholar