American Institute of Mathematical Sciences

Advanced
  2017, 11: 43-56. doi: 10.3934/jmd.2017003

Positive metric entropy in nondegenerate nearly integrable systems

Dong Chen

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received  May 25, 2016 Revised  October 06, 2016 Published  December 2016

Fund Project: The author is supported by Dmitri Burago's department research fund 42844-1001.

The celebrated KAM theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^\infty$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori.

Keywords: KAM theory, Finsler metric, dual lens map, Hamiltonian flow, perturbation, metric entropy.
Mathematics Subject Classification: Primary: 37A35, 37J40; Secondary: 53C60.
Citation: Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003
References:
[1]

V. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math. Survey, 18 (1963), 9-36.

[2]

V. Arnol'd, Instability of dynamical systems with several degrees of freedom, Soviet Mathematics, 5 (1964), 581-585. 

[3]

A. Bolsinov and I. Ta${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$manov, Integrable geodesic flows on suspensions of automorphisms of tori, Proc. Steklov Institute Math, 231 (2000), 42-58. 

[4]

D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics, Geom. Topol., 20 (2016), 469-490.  doi: 10.2140/gt.2016.20.469.

[5]

K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on S2, Ergodic Theory Dynam. Systems, 9 (1989), 27-45.  doi: 10.1017/S0143385700004806.

[6]

G. Contreras, Geodesic flows with positive topological entropy, twist map and hyperbolicity. (2), Ann. of Math, 172 (2010), 761-808.  doi: 10.4007/annals.2010.172.761.

[7]

V. Donnay, Geodesic flow on the two-sphere. Ⅰ. Positive measure entropy, Ergodic Theory Dynam. Systems, 8 (1988), 531-553.  doi: 10.1017/S0143385700004685.

[8]

V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic, Comm. Math. Phys, 135 (1991), 267-302.  doi: 10.1007/BF02098044.

[9]

F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8,1948, Interscience Publishers, Inc., New York, N. Y., 1948,187–204.

[10]

A. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 525-530. 

[11]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1962 (1962), 1-20. 

[12]

N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Functional Analysis and Its Applications, 5 (1971), 338-339.  doi: 10.1007/BF01086753.

[13]

S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Amer. J.Math., 99 (1977), 1061-1087.  doi: 10.2307/2374000.

[14]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.

[15]

Ja. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspekhi Mat. Nauk, 32 (1977), 55-287. 

[16]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668.

[17]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam.Systems, 5 (1985), 145-161.  doi: 10.1017/S0143385700002807.

show all references

References:
[1]

V. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russian Math. Survey, 18 (1963), 9-36.

[2]

V. Arnol'd, Instability of dynamical systems with several degrees of freedom, Soviet Mathematics, 5 (1964), 581-585. 

[3]

A. Bolsinov and I. Ta${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$manov, Integrable geodesic flows on suspensions of automorphisms of tori, Proc. Steklov Institute Math, 231 (2000), 42-58. 

[4]

D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics, Geom. Topol., 20 (2016), 469-490.  doi: 10.2140/gt.2016.20.469.

[5]

K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on S2, Ergodic Theory Dynam. Systems, 9 (1989), 27-45.  doi: 10.1017/S0143385700004806.

[6]

G. Contreras, Geodesic flows with positive topological entropy, twist map and hyperbolicity. (2), Ann. of Math, 172 (2010), 761-808.  doi: 10.4007/annals.2010.172.761.

[7]

V. Donnay, Geodesic flow on the two-sphere. Ⅰ. Positive measure entropy, Ergodic Theory Dynam. Systems, 8 (1988), 531-553.  doi: 10.1017/S0143385700004685.

[8]

V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic, Comm. Math. Phys, 135 (1991), 267-302.  doi: 10.1007/BF02098044.

[9]

F. John, Extremum problems with inequalities as subsidiary conditions, in Studies and Essays Presented to R. Courant on his 60th Birthday, January 8,1948, Interscience Publishers, Inc., New York, N. Y., 1948,187–204.

[10]

A. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 525-530. 

[11]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. Ⅱ, 1962 (1962), 1-20. 

[12]

N. Nekhoroshev, Behavior of Hamiltonian systems close to integrable, Functional Analysis and Its Applications, 5 (1971), 338-339.  doi: 10.1007/BF01086753.

[13]

S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Amer. J.Math., 99 (1977), 1061-1087.  doi: 10.2307/2374000.

[14]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.

[15]

Ja. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspekhi Mat. Nauk, 32 (1977), 55-287. 

[16]

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tôhoku Math. J. (2), 10 (1958), 338-354.  doi: 10.2748/tmj/1178244668.

[17]

M. Wojtkowski, Invariant families of cones and Lyapunov exponents, Ergodic Theory Dynam.Systems, 5 (1985), 145-161.  doi: 10.1017/S0143385700002807.

Figure 1.  A non-ergodic DBG torus
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Graphs of $u_S$, $u_C$ and $u$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Graph of $\rho$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Construction of $\phi_1$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xufeng Guo, Gang Liao, Wenxiang Sun, Dawei Yang. On the hybrid control of metric entropy for dominated splittings. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5011-5019. doi: 10.3934/dcds.2018219
[2]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[3]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of volume preserving Anosov systems. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4767-4783. doi: 10.3934/dcds.2017205
[4]

Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (11) : 6589-6604. doi: 10.3934/dcdsb.2022010
[5]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437
[6]

Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems and Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035
[7]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[8]

Jintao Wang, Desheng Li, Jinqiao Duan. On the shape Conley index theory of semiflows on complete metric spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1629-1647. doi: 10.3934/dcds.2016.36.1629
[9]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[10]

Matthias Erbar, Dominik Forkert, Jan Maas, Delio Mugnolo. Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph. Networks and Heterogeneous Media, 2022, 17 (5) : 687-717. doi: 10.3934/nhm.2022023
[11]

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 and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
[12]

Mourad Bellassoued, Zouhour Rezig. Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1061-1084. doi: 10.3934/dcdss.2021158
[13]

Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010
[14]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[15]

Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299
[16]

Vincenzo Recupero. Hysteresis operators in metric spaces. Discrete and Continuous Dynamical Systems - S, 2015, 8 (4) : 773-792. doi: 10.3934/dcdss.2015.8.773
[17]

Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443
[18]

Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure and Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103
[19]

Peter Giesl, Holger Wendland. Construction of a contraction metric by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3843-3863. doi: 10.3934/dcdsb.2018333
[20]

Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733

2021 Impact Factor: 0.641

Tools

Metrics

  • PDF downloads (96)
  • HTML views (84)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy