# American Institute of Mathematical Sciences

2017, 11: 425-445. doi: 10.3934/jmd.2017017

## An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms

 IMJ-PRG, UP7D 58-56 Avenue de France, 75205 Paris Cedex 13, France

Received  January 02, 2017 Revised  April 19, 2017 Published  November 2017

Fund Project: BF: Supported in part by Projet ANR-15-CE40-0001.

For area preserving C2 surface diffeomorphisms, we give an explicit finite information condition on the exponential growth of the number of Bowen's (n, δ)-balls needed to cover a positive proportion of the space, that is sufficient to guarantee positive topological entropy. This can be seen as an effective version of Katok's horseshoe theorem in the conservative setting. We also show that the analogous result is false in dimension larger than 3.

Citation: Bassam Fayad, Zhiyuan Zhang. An effective version of Katok's horseshoe theorem for conservative C2 surface diffeomorphisms. Journal of Modern Dynamics, 2017, 11: 425-445. doi: 10.3934/jmd.2017017
##### References:
 [1] L. M. Abramov and V. A. Rohlin, The entropy of a skew product of measure preserving transformations, AMS Translations, 48 (1965), 255-265.  doi: 10.1090/trans2/048.  Google Scholar [2] A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, On mixing diffeomorphisms of the disk, arXiv: 1509.06906. Google Scholar [3] V. Climenhaga, D. Dolgopyat and Y. Pesin, Non-stationary non-uniform hyperbolicity: SRB measures for non-uniformly hyperbolic attractors, Comm. Math. Phys., 346 (2016), 553-602.  doi: 10.1007/s00220-016-2710-z.  Google Scholar [4] V. Climenhaga and Y. Pesin, Hadamard-Perron theorems and effective hyperbolicity, Ergodic Theory Dynam. Systems, 36 (2016), 23-63.  doi: 10.1017/etds.2014.49.  Google Scholar [5] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137-173.   Google Scholar [6] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [7] V. Pliss, On a conjecture of Smale, Diff, Uravnenjia, 8 (1972), 268-282.   Google Scholar

show all references

##### References:
 [1] L. M. Abramov and V. A. Rohlin, The entropy of a skew product of measure preserving transformations, AMS Translations, 48 (1965), 255-265.  doi: 10.1090/trans2/048.  Google Scholar [2] A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, On mixing diffeomorphisms of the disk, arXiv: 1509.06906. Google Scholar [3] V. Climenhaga, D. Dolgopyat and Y. Pesin, Non-stationary non-uniform hyperbolicity: SRB measures for non-uniformly hyperbolic attractors, Comm. Math. Phys., 346 (2016), 553-602.  doi: 10.1007/s00220-016-2710-z.  Google Scholar [4] V. Climenhaga and Y. Pesin, Hadamard-Perron theorems and effective hyperbolicity, Ergodic Theory Dynam. Systems, 36 (2016), 23-63.  doi: 10.1017/etds.2014.49.  Google Scholar [5] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137-173.   Google Scholar [6] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar [7] V. Pliss, On a conjecture of Smale, Diff, Uravnenjia, 8 (1972), 268-282.   Google Scholar
$\mathcal{R}_1$ is the topological rectangle $abcd$; $\mathcal{R}_2$ is the topological rectangle $a'b'c'd'$. Under a hyperbolic map $G$, $ab$ is mapped to $a'b'$ and similarly $bc, cd, da$ are mapped respectively to $b'c', c'd', d'a'$.
 [1] Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421 [2] Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195 [3] Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123 [4] Christian Bonatti, Sylvain Crovisier, Katsutoshi Shinohara. The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited. Journal of Modern Dynamics, 2013, 7 (4) : 605-618. doi: 10.3934/jmd.2013.7.605 [5] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [6] Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 589-607. doi: 10.3934/dcds.2010.27.589 [7] John Hubbard, Yulij Ilyashenko. A proof of Kolmogorov's theorem. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 367-385. doi: 10.3934/dcds.2004.10.367 [8] Rabah Amir, Igor V. Evstigneev. On Zermelo's theorem. Journal of Dynamics & Games, 2017, 4 (3) : 191-194. doi: 10.3934/jdg.2017011 [9] Guillaume Bal. Homogenization in random media and effective medium theory for high frequency waves. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 473-492. doi: 10.3934/dcdsb.2007.8.473 [10] Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313 [11] Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109 [12] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [13] Fabio Cipriani, Gabriele Grillo. On the $l^p$ -agmon's theory. Conference Publications, 1998, 1998 (Special) : 167-176. doi: 10.3934/proc.1998.1998.167 [14] Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427 [15] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [16] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 [17] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [18] Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945 [19] Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43 [20] Fatiha Alabau-Boussouira, Piermarco Cannarsa. A constructive proof of Gibson's stability theorem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 611-617. doi: 10.3934/dcdss.2013.6.611

2020 Impact Factor: 0.848