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January  2020, 40(1): 441-465. doi: 10.3934/dcds.2020017

Anosov diffeomorphism with a horseshoe that attracts almost any point

Christian Bonatti 1, , Stanislav Minkov 2, , Alexey Okunev 3, and  Ivan Shilin 4,

1. 

Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
2. 

Brook Institute of Electronic Control Machines, 119334, Moscow, Vavilova str., 24, Russia
3. 

Loughborough University, LE11 3TU, Loughborough, UK
4. 

National Research University Higher School of Economics, Faculty of Mathematics, 119048, Moscow, Usacheva str., 6, Russia

Received  March 2019 Revised  June 2019 Published  October 2019

Fund Project: Supported in part by the RFBR grant 16-01-00748-a

We present an example of a $ C^1 $ Anosov diffeomorphism of a two-torus with a physical measure such that its basin has full Lebesgue measure and its support is a horseshoe of zero measure.

Keywords: Anosov diffeomorphisms, physical measure, Milnor attractor, C1-dynamics, thick Cantor sets.
Mathematics Subject Classification: Primary: 37D20; Secondary: 37C40, 37C70, 37E30, 37G35.
Citation: Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 441-465. doi: 10.3934/dcds.2020017
References:
[1]

C. Bonatti, L. J. Daz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics III), Springer, Berlin, 2005. doi: 10.1007/b138174.  Google Scholar

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Yu. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, Nonlinearity, 25 (2012), 2377-2399.  doi: 10.1088/0951-7715/25/8/2377.  Google Scholar

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I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc., 31 (1994), 68-74.  doi: 10.1090/S0273-0979-1994-00507-5.  Google Scholar

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V. Kleptsyn, D. Ryzhov and S. Minkov, Special ergodic theorems and dynamical large deviations, Nonlinearity, 25 (2012), 3189-3196.  doi: 10.1088/0951-7715/25/11/3189.  Google Scholar

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V. Kleptsyn and P. Saltykov, On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps, Trans. Moscow Math. Soc., 72 (2011), 193-217.  doi: 10.1090/s0077-1554-2012-00196-4.  Google Scholar

[8]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.  Google Scholar

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S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.  doi: 10.2307/2373372.  Google Scholar

[10]

C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176.  doi: 10.1007/BF01390119.  Google Scholar

[11]

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

[12]

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

show all references

References:
[1]

C. Bonatti, L. J. Daz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics III), Springer, Berlin, 2005. doi: 10.1007/b138174.  Google Scholar

[2]

R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.  doi: 10.1007/BF01389849.  Google Scholar

[3]

R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-77695-6.  Google Scholar

[4]

Yu. Ilyashenko and A. Negut, Hölder properties of perturbed skew products and Fubini regained, Nonlinearity, 25 (2012), 2377-2399.  doi: 10.1088/0951-7715/25/8/2377.  Google Scholar

[5]

I. Kan, Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin, Bull. Amer. Math. Soc., 31 (1994), 68-74.  doi: 10.1090/S0273-0979-1994-00507-5.  Google Scholar

[6]

V. Kleptsyn, D. Ryzhov and S. Minkov, Special ergodic theorems and dynamical large deviations, Nonlinearity, 25 (2012), 3189-3196.  doi: 10.1088/0951-7715/25/11/3189.  Google Scholar

[7]

V. Kleptsyn and P. Saltykov, On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps, Trans. Moscow Math. Soc., 72 (2011), 193-217.  doi: 10.1090/s0077-1554-2012-00196-4.  Google Scholar

[8]

J. Milnor, On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.  doi: 10.1007/BF01212280.  Google Scholar

[9]

S. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.  doi: 10.2307/2373372.  Google Scholar

[10]

C. Robinson and L. S. Young, Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176.  doi: 10.1007/BF01390119.  Google Scholar

[11]

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

[12]

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

Figure 1.  The set $ UK $. The vertical stripes of $ UK \setminus K $ are depicted wider than they are.
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Figure 2.  The "frame" $ R $.
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Figure 3.  $ \tilde R $, $ UH_i $, $ \tilde RH_i $, $ RH_i $.
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