az and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective (Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics III), Springer, Berlin, 2005.
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The set $ UK $ . The vertical stripes of $ UK \setminus K $ are depicted wider than they are.
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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
| 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 |
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 and Continuous Dynamical Systems,
2020, 40
(1)
: 441-465.
doi: 10.3934/dcds.2020017
![]()
References:
| [1] |
C. Bonatti, L. J. D |
| [2] |
R. Bowen,
A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.
doi: 10.1007/BF01389849. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
J. Milnor,
On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.
doi: 10.1007/BF01212280. |
| [9] |
S. Newhouse,
On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
| [10] |
C. Robinson and L. S. Young,
Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176.
doi: 10.1007/BF01390119. |
| [11] |
D. Ruelle,
A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [12] |
Ya. Sinai,
Gibbs measures in ergodic theory, Russian Math. Surveys., 27 (1972), 21-64.
|
show all references
References:
| [1] |
C. Bonatti, L. J. D |
| [2] |
R. Bowen,
A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.
doi: 10.1007/BF01389849. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
J. Milnor,
On the concept of attractor, Comm. Math. Phys., 99 (1985), 177-195.
doi: 10.1007/BF01212280. |
| [9] |
S. Newhouse,
On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 (1970), 761-770.
doi: 10.2307/2373372. |
| [10] |
C. Robinson and L. S. Young,
Nonabsolutely continuous foliations for an Anosov diffeomorphism, Invent. Math., 61 (1980), 159-176.
doi: 10.1007/BF01390119. |
| [11] |
D. Ruelle,
A measure associated with Axiom A attractors, Amer. J. Math., 98 (1976), 619-654.
doi: 10.2307/2373810. |
| [12] |
Ya. Sinai,
Gibbs measures in ergodic theory, Russian Math. Surveys., 27 (1972), 21-64.
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