A robustly transitive diffeomorphism of Kan's type

Cheng Cheng; Shaobo Gan; Yi Shi

doi:10.3934/dcds.2018037

Article Contents

2018, Volume 38, Issue 2: 867-888. Doi: 10.3934/dcds.2018037

This issue Previous Article Reversing and extended symmetries of shift spaces Next Article Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

A robustly transitive diffeomorphism of Kan's type

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

Received: December 31, 2016

Revised: July 31, 2017

Published: November 2017

The second author is supported by NSFC grant 11231001

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D30; Secondary: 37D25, 37D35.

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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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References

	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.
	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.
	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.
	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.
	C. Bonatti and L. J. Díaz , Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math.(2), 143 (1996) , 357-396. doi: 10.2307/2118647.
	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.
	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.
	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.
	C. Bonatti , L. J. Díaz and M. Viana , Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia Math. Sci., Springer-Verlag, (2005) .
	C. Bonatti and R. Potrie, Many intermingled basins in dimension 3, preprint, arXiv: 1603.03803v1.
	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.
	R. Bowen and D. Ruelle , The ergodic theory of Axiom A flows, Invent. Math., 29 (1975) , 181-202. doi: 10.1007/BF01389848.
	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.
	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.
	S. Gan and Y. Shi, Topological mixing for Kan's map, In preparation.
	M. Hirsch , C. Pugh and M. Shub , Invariant Manifolds, Lecture Notes in Mathmatics, Springer-Verlag, Berlin, (1977) .
	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.
	J. E. Hutchinson , Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981) , 713-747. doi: 10.1512/iumj.1981.30.30055.
	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.
	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.
	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.
	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.
	M. Nassiri and E. Pujals , Robust transitivity in Hamiltonian dynamics, Ann. Sci. Ãƒâ€°c. Norm. SupÃƒ©r., 45 (2012) , 191-239. doi: 10.24033/asens.2164.
	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.
	J. Palis , A global perspective for non-conservative dynamics, Ann. Inst. H. PoincarÃƒ© Anal. Non LinÃƒ©aire, 22 (2005) , 485-507. doi: 10.1016/j.anihpc.2005.01.001.
	Y. Pesin and Y. Sinai , Gibbs measures for partially hyperbolic attractors, Ergod. Th. & Dynam. Sys., 2 (1982) , 417-438. doi: 10.1017/S014338570000170X.
	D. Ruelle , A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976) , 619-654. doi: 10.2307/2373810.
	Ya. Sinai , Gibbs measures in ergodic theory, Russian Math. Surveys, 27 (1972) , 21-64.
	R. Ures and C. H. Vasquez, On the robustness of intermingled basins preprint, arXiv: 1503.07155v2. doi: 10.1017/etds.2016.33.
	M. Viana and J. Yang , Physical measures and absolute continuity for one-dimensional center direction, Ann. Inst. H. PoincarÃƒ© Anal. Non LinÃƒ©aire, 30 (2013) , 845-877. doi: 10.1016/j.anihpc.2012.11.002.
	J. Yang, Entropy along expanding foliations, preprint, arXiv: 1601.05504v1.