# American Institute of Mathematical Sciences

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

## Anosov diffeomorphism with a horseshoe that attracts almost any point

 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.

Citation: Christian Bonatti, Stanislav Minkov, Alexey Okunev, Ivan Shilin. Anosov diffeomorphism with a horseshoe that attracts almost any point. Discrete & Continuous Dynamical Systems, 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 [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

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