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doi: 10.3934/dcds.2022052
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An upper bound of the measure-theoretical entropy

Soochow College, Soochow University, Suzhou, 215006, China

*Corresponding author: Yuntao Zang

Received  September 2021 Revised  February 2022 Early access April 2022

Fund Project: The author is supported by China Postdoctoral Science Foundation (2020TQ0098, 2021M690057)

Let $ f $ be a $ C^{1+\alpha} $ diffeomorphism on a compact manifold $ M $ and let $ \mu $ be an ergodic measure. We use a special family of fake center-stable manifolds to bound the entropy of $ \mu $ in terms of positive Lyapunov exponents and the so called 'dimensional entropy', a notion related to the topological entropy of submanifolds.

Citation: Yuntao Zang. An upper bound of the measure-theoretical entropy. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022052
References:
[1]

J. Buzzi, Dimensional entropies and semi-uniform hyperbolicity, In New Trends in Mathematical Physics, (2009), 95–116. doi: 10.1007/978-90-481-2810-5_8.

[2]

K. Cogswell, Entropy and volume growth, Ergodic Theory Dynam. Systems, 20 (2000), 77-84.  doi: 10.1017/S0143385700000055.

[3]

A. Fathi, M.-R. Herman and J.-C. Yoccoz, A proof of Pesin's stable manifold theorem, Geometric Dynamics Lecture Notes in Math., Springer, Berlin, 1007 (1983), 177–215. doi: 10.1007/BFb0061417.

[4]

X. GuoG. LiaoW. Sun and D. Yang, On the hybrid control of metric entropy for dominated splittings, Discrete Contin. Dyn. Syst., 38 (2018), 5011-5019.  doi: 10.3934/dcds.2018219.

[5]

Y. HuaR. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 28 (2008), 843-862.  doi: 10.1017/S0143385707000405.

[6]

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

[7]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861-864. 

[8]

W. Kulpa, The Poincaré-Miranda theorem, Amer. Math. Monthly, 104 (1997), 545-550.  doi: 10.2307/2975081.

[9]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574.  doi: 10.2307/1971329.

[10]

S. E. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8 (1988), 283-299.  doi: 10.1017/S0143385700009469.

[11]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obš č., 19 (1968), 179-210. 

[12]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.

[13]

R. Saghin, Volume growth and entropy for $C^{1}$ partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 34 (2014), 3789-3801.  doi: 10.3934/dcds.2014.34.3789.

[14]

P. Walters, An Introduction to Ergodic Theory, Volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[15]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.

[16]

Y. Zang, Entropies and volume growth of strong unstable manifolds, Ergodic Theory Dynam. Systems, 42 (2022), 1576-1590.  doi: 10.1017/etds.2021.2.

show all references

References:
[1]

J. Buzzi, Dimensional entropies and semi-uniform hyperbolicity, In New Trends in Mathematical Physics, (2009), 95–116. doi: 10.1007/978-90-481-2810-5_8.

[2]

K. Cogswell, Entropy and volume growth, Ergodic Theory Dynam. Systems, 20 (2000), 77-84.  doi: 10.1017/S0143385700000055.

[3]

A. Fathi, M.-R. Herman and J.-C. Yoccoz, A proof of Pesin's stable manifold theorem, Geometric Dynamics Lecture Notes in Math., Springer, Berlin, 1007 (1983), 177–215. doi: 10.1007/BFb0061417.

[4]

X. GuoG. LiaoW. Sun and D. Yang, On the hybrid control of metric entropy for dominated splittings, Discrete Contin. Dyn. Syst., 38 (2018), 5011-5019.  doi: 10.3934/dcds.2018219.

[5]

Y. HuaR. Saghin and Z. Xia, Topological entropy and partially hyperbolic diffeomorphisms, Ergodic Theory Dynam. Systems, 28 (2008), 843-862.  doi: 10.1017/S0143385707000405.

[6]

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

[7]

A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.), 119 (1958), 861-864. 

[8]

W. Kulpa, The Poincaré-Miranda theorem, Amer. Math. Monthly, 104 (1997), 545-550.  doi: 10.2307/2975081.

[9]

F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math., 122 (1985), 540-574.  doi: 10.2307/1971329.

[10]

S. E. Newhouse, Entropy and volume, Ergodic Theory Dynam. Systems, 8 (1988), 283-299.  doi: 10.1017/S0143385700009469.

[11]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obš č., 19 (1968), 179-210. 

[12]

D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat., 9 (1978), 83-87.  doi: 10.1007/BF02584795.

[13]

R. Saghin, Volume growth and entropy for $C^{1}$ partially hyperbolic diffeomorphisms, Discrete Contin. Dyn. Syst., 34 (2014), 3789-3801.  doi: 10.3934/dcds.2014.34.3789.

[14]

P. Walters, An Introduction to Ergodic Theory, Volume 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[15]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.

[16]

Y. Zang, Entropies and volume growth of strong unstable manifolds, Ergodic Theory Dynam. Systems, 42 (2022), 1576-1590.  doi: 10.1017/etds.2021.2.

Figure 1.  Standard family
Figure 2.  Maximal separated subset of $ \Lambda $
Figure 3.  Second property in Lemma 2.4
Figure 4.  Expansion on different directions
Figure 5.  Geometric estimate
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