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February  2018, 38(2): 867-888. doi: 10.3934/dcds.2018037

A robustly transitive diffeomorphism of Kan's type

Cheng Cheng , Shaobo Gan and  Yi Shi

School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  January 2017 Revised  August 2017 Published  February 2018

Fund Project: The second author is supported by NSFC grant 11231001.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396.  doi: 10.2307/2118647.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., Springer-Verlag, (2005).   Google Scholar

[10]

C. Bonatti and R. Potrie, Many intermingled basins in dimension 3, preprint, arXiv: 1603.03803v1. Google Scholar

[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.  Google Scholar

[12]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Gan and Y. Shi, Topological mixing for Kan's map, In preparation. Google Scholar

[16]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathmatics, Springer-Verlag, Berlin, (1977).   Google Scholar

[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.  Google Scholar

[18]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[27]

D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[28]

Ya. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.   Google Scholar

[29]

R. Ures and C. H. Vasquez, On the robustness of intermingled basins preprint, arXiv: 1503.07155v2. doi: 10.1017/etds.2016.33.  Google Scholar

[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.  Google Scholar

[31]

J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504v1. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996), 357-396.  doi: 10.2307/2118647.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., Springer-Verlag, (2005).   Google Scholar

[10]

C. Bonatti and R. Potrie, Many intermingled basins in dimension 3, preprint, arXiv: 1603.03803v1. Google Scholar

[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.  Google Scholar

[12]

R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math., 29 (1975), 181-202.  doi: 10.1007/BF01389848.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

S. Gan and Y. Shi, Topological mixing for Kan's map, In preparation. Google Scholar

[16]

M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathmatics, Springer-Verlag, Berlin, (1977).   Google Scholar

[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.  Google Scholar

[18]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Y. Pesin and Y. Sinai, Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982), 417-438.  doi: 10.1017/S014338570000170X.  Google Scholar

[27]

D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.  Google Scholar

[28]

Ya. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972), 21-64.   Google Scholar

[29]

R. Ures and C. H. Vasquez, On the robustness of intermingled basins preprint, arXiv: 1503.07155v2. doi: 10.1017/etds.2016.33.  Google Scholar

[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.  Google Scholar

[31]

J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504v1. Google Scholar

Figure 1.  Perturbation domains.
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Figure 2.  Construction of blender-horseshoe.
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Figure 3.  Robustly transitive Kan's example on $\mathbb{T}^3$.
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Figure 4.  ${\rm{Supp}}(\mu_1)$ is $s$-saturated.
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