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

Positive metric entropy in nondegenerate nearly integrable systems

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.

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.Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[17]

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

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.Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[17]

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

Figure 1.  A non-ergodic DBG torus
Figure 2.  Graphs of $u_S$, $u_C$ and $u$
Figure 3.  Graph of $\rho$
Figure 4.  Construction of $\phi_1$
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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.

[8]

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 - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641

[18]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[19]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[20]

Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030

2018 Impact Factor: 0.295

Metrics

  • PDF downloads (15)
  • HTML views (37)
  • Cited by (0)

Other articles
by authors

[Back to Top]