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February
2018, 38(2): 867-888.
doi: 10.3934/dcds.2018037
A robustly transitive diffeomorphism of Kan's type
School of Mathematical Sciences, Peking University, Beijing 100871, China |
We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admit two physical measures with intermingled basins. In particular, all these diffeomorphisms are not topologically mixing. Moreover, every such example exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.
Keywords:
Robustly transitive,
physical measure,
intermingled basin,
Gibbs state,
blender-horseshoe.
Mathematics Subject Classification:
Primary: 37D30; Secondary: 37D25, 37D35.
Citation:
Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete and Continuous Dynamical Systems,
2018, 38
(2)
: 867-888.
doi: 10.3934/dcds.2018037
![]()
References:
| [1] |
F. Abdenur and S. Crovisier,
Transitivity and topological mixing for $C^1$ diffeomorphisms, in Essays in Mathematics and Its Applications, Springer, Heidelberg, (2012), 1-16.
doi: 10.1007/978-3-642-28821-0_1. |
| [2] |
J. F. Alves, C. Bonatti and M. Viana,
SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.
doi: 10.1007/s002220000057. |
| [3] |
M. Anderson,
Robust ergodic properties in partially hyperbolic dynamics, Trans. Amer. Math. Soc., 362 (2010), 1831-1867.
doi: 10.1090/S0002-9947-09-05027-2. |
| [4] |
P. G. Barrientos, Y. Ki and A. Raibekas,
Symbolic blender-horseshoes and applications, Nonlinearity, 27 (2014), 2805-2839.
doi: 10.1088/0951-7715/27/12/2805. |
| [5] |
C. Bonatti and L. J. Díaz,
Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396.
doi: 10.2307/2118647. |
| [6] |
C. Bonatti and L. J. Díaz,
Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525.
doi: 10.1017/S1474748008000030. |
| [7] |
C. Bonatti and L. J. Díaz,
Abundance of $C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148.
doi: 10.1090/S0002-9947-2012-05445-6. |
| [8] |
C. Bonatti, L. J. Díaz and R. Ures,
Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.
doi: 10.1017/S1474748002000142. |
| [9] |
C. Bonatti, L. J. Díaz and M. Viana,
Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., Springer-Verlag, (2005).
|
| [10] |
C. Bonatti and R. Potrie, Many intermingled basins in dimension 3, preprint, arXiv: 1603.03803v1. |
| [11] |
C. Bonatti and M. Viana,
SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting, Isreal J. of Math., 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
| [12] |
R. Bowen and D. Ruelle,
The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.
doi: 10.1007/BF01389848. |
| [13] |
K. Burns, F. R. Hertz, J. R. Hertz, A. Talitskaya and R. Ures,
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
| [14] |
D. Dolgopyat, M. Viana and J. Yang,
Geometric and measure-theoretical structures of maps with mostly contracting center, Commun. Math. Phys., 341 (2016), 991-1014.
doi: 10.1007/s00220-015-2554-y. |
| [15] |
S. Gan and Y. Shi, Topological mixing for Kan's map, In preparation. |
| [16] |
M. Hirsch, C. Pugh and M. Shub,
Invariant Manifolds, Lecture Notes in Mathmatics, Springer-Verlag, Berlin, (1977).
|
| [17] |
A. J. Homburg and M. Nassiri,
Robust minimality of iterated function systems with two generators, Ergod. Th. & Dynam. Sys., 34 (2014), 1914-1929.
doi: 10.1017/etds.2013.34. |
| [18] |
J. E. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
| [19] |
Y. S. Ilyashenko, V. A. Kleptsyn and P. Saltykov,
Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, J. Fixed Point Theory Appl., 3 (2008), 449-463.
doi: 10.1007/s11784-008-0088-z. |
| [20] |
I. Kan,
Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc. (N.S.), 31 (1994), 68-74.
doi: 10.1090/S0273-0979-1994-00507-5. |
| [21] |
V. A. Kleptsyn and P. S. Saltykov,
On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps, Trans. Moscow Math. Soc., (2011), 193-217.
doi: 10.1090/s0077-1554-2012-00196-4. |
| [22] |
S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke,
Fractal basin boundaries, Phys. D., 17 (1985), 125-153.
doi: 10.1016/0167-2789(85)90001-6. |
| [23] |
M. Nassiri and E. Pujals,
Robust transitivity in Hamiltonian dynamics, Ann. Sci. Éc. Norm. Supér., 45 (2012), 191-239.
doi: 10.24033/asens.2164. |
| [24] |
A. Okunev,
Milnor attractors of skew products with the fiber a circle, J. Dyn. Control Syst., 23 (2017), 421-433.
doi: 10.1007/s10883-016-9334-7. |
| [25] |
J. Palis,
A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507.
doi: 10.1016/j.anihpc.2005.01.001. |
| [26] |
Y. Pesin and Y. Sinai,
Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.
doi: 10.1017/S014338570000170X. |
| [27] |
D. Ruelle,
A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [28] |
Ya. Sinai,
Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.
|
| [29] |
R. Ures and C. H. Vasquez, On the robustness of intermingled basins preprint, arXiv: 1503.07155v2.
doi: 10.1017/etds.2016.33. |
| [30] |
M. Viana and J. Yang,
Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 845-877.
doi: 10.1016/j.anihpc.2012.11.002. |
| [31] |
J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504v1. |
show all references
References:
| [1] |
F. Abdenur and S. Crovisier,
Transitivity and topological mixing for $C^1$ diffeomorphisms, in Essays in Mathematics and Its Applications, Springer, Heidelberg, (2012), 1-16.
doi: 10.1007/978-3-642-28821-0_1. |
| [2] |
J. F. Alves, C. Bonatti and M. Viana,
SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.
doi: 10.1007/s002220000057. |
| [3] |
M. Anderson,
Robust ergodic properties in partially hyperbolic dynamics, Trans. Amer. Math. Soc., 362 (2010), 1831-1867.
doi: 10.1090/S0002-9947-09-05027-2. |
| [4] |
P. G. Barrientos, Y. Ki and A. Raibekas,
Symbolic blender-horseshoes and applications, Nonlinearity, 27 (2014), 2805-2839.
doi: 10.1088/0951-7715/27/12/2805. |
| [5] |
C. Bonatti and L. J. Díaz,
Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396.
doi: 10.2307/2118647. |
| [6] |
C. Bonatti and L. J. Díaz,
Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525.
doi: 10.1017/S1474748008000030. |
| [7] |
C. Bonatti and L. J. Díaz,
Abundance of $C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc., 364 (2012), 5111-5148.
doi: 10.1090/S0002-9947-2012-05445-6. |
| [8] |
C. Bonatti, L. J. Díaz and R. Ures,
Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.
doi: 10.1017/S1474748002000142. |
| [9] |
C. Bonatti, L. J. Díaz and M. Viana,
Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., Springer-Verlag, (2005).
|
| [10] |
C. Bonatti and R. Potrie, Many intermingled basins in dimension 3, preprint, arXiv: 1603.03803v1. |
| [11] |
C. Bonatti and M. Viana,
SRB measures for partially hyperbolic diffeomorphisms whose central direction is mostly contracting, Isreal J. of Math., 115 (2000), 157-193.
doi: 10.1007/BF02810585. |
| [12] |
R. Bowen and D. Ruelle,
The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.
doi: 10.1007/BF01389848. |
| [13] |
K. Burns, F. R. Hertz, J. R. Hertz, A. Talitskaya and R. Ures,
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete Contin. Dyn. Syst., 22 (2008), 75-88.
doi: 10.3934/dcds.2008.22.75. |
| [14] |
D. Dolgopyat, M. Viana and J. Yang,
Geometric and measure-theoretical structures of maps with mostly contracting center, Commun. Math. Phys., 341 (2016), 991-1014.
doi: 10.1007/s00220-015-2554-y. |
| [15] |
S. Gan and Y. Shi, Topological mixing for Kan's map, In preparation. |
| [16] |
M. Hirsch, C. Pugh and M. Shub,
Invariant Manifolds, Lecture Notes in Mathmatics, Springer-Verlag, Berlin, (1977).
|
| [17] |
A. J. Homburg and M. Nassiri,
Robust minimality of iterated function systems with two generators, Ergod. Th. & Dynam. Sys., 34 (2014), 1914-1929.
doi: 10.1017/etds.2013.34. |
| [18] |
J. E. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
| [19] |
Y. S. Ilyashenko, V. A. Kleptsyn and P. Saltykov,
Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins, J. Fixed Point Theory Appl., 3 (2008), 449-463.
doi: 10.1007/s11784-008-0088-z. |
| [20] |
I. Kan,
Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc. (N.S.), 31 (1994), 68-74.
doi: 10.1090/S0273-0979-1994-00507-5. |
| [21] |
V. A. Kleptsyn and P. S. Saltykov,
On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps, Trans. Moscow Math. Soc., (2011), 193-217.
doi: 10.1090/s0077-1554-2012-00196-4. |
| [22] |
S. W. McDonald, C. Grebogi, E. Ott and J. A. Yorke,
Fractal basin boundaries, Phys. D., 17 (1985), 125-153.
doi: 10.1016/0167-2789(85)90001-6. |
| [23] |
M. Nassiri and E. Pujals,
Robust transitivity in Hamiltonian dynamics, Ann. Sci. Éc. Norm. Supér., 45 (2012), 191-239.
doi: 10.24033/asens.2164. |
| [24] |
A. Okunev,
Milnor attractors of skew products with the fiber a circle, J. Dyn. Control Syst., 23 (2017), 421-433.
doi: 10.1007/s10883-016-9334-7. |
| [25] |
J. Palis,
A global perspective for non-conservative dynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 485-507.
doi: 10.1016/j.anihpc.2005.01.001. |
| [26] |
Y. Pesin and Y. Sinai,
Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.
doi: 10.1017/S014338570000170X. |
| [27] |
D. Ruelle,
A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [28] |
Ya. Sinai,
Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.
|
| [29] |
R. Ures and C. H. Vasquez, On the robustness of intermingled basins preprint, arXiv: 1503.07155v2.
doi: 10.1017/etds.2016.33. |
| [30] |
M. Viana and J. Yang,
Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 845-877.
doi: 10.1016/j.anihpc.2012.11.002. |
| [31] |
J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504v1. |
Figure 2.
Construction of blender-horseshoe.
Figure 3.
Robustly transitive Kan's example on $\mathbb{T}^3$ .
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