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.

[2]

A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, On mixing diffeomorphisms of the disk, arXiv: 1509.06906.

[3]

V. ClimenhagaD. 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.

[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.

[5]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137-173. 

[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.

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V. Pliss, On a conjecture of Smale, Diff, Uravnenjia, 8 (1972), 268-282. 

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.

[2]

A. Avila, B. Fayad, P. Le Calvez, D. Xu and Z. Zhang, On mixing diffeomorphisms of the disk, arXiv: 1509.06906.

[3]

V. ClimenhagaD. 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.

[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.

[5]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes études Sci. Publ. Math., 51 (1980), 137-173. 

[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.

[7]

V. Pliss, On a conjecture of Smale, Diff, Uravnenjia, 8 (1972), 268-282. 

Figure 1.  $\mathcal{R}_1$ 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 $b'c', c'd', d'a'$.
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