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.

[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 2.  Graphs of $u_S$, $u_C$ and $u$
Figure 3.  Graph of $\rho$
Figure 4.  Construction of $\phi_1$
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