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doi: 10.3934/dcds.2021164
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Quantitative destruction of invariant circles

Lin Wang

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Received  March 2021 Revised  August 2021 Early access November 2021

For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency $ \omega $ of an integrable system by a trigonometric polynomial of degree $ N $ perturbation $ R_N $ with $ \|R_N\|_{C^r}<\epsilon $. We obtain a relation among $ N $, $ r $, $ \epsilon $ and the arithmetic property of $ \omega $, for which the area-preserving map admit no invariant circles with $ \omega $.

Keywords: Invariant circle, minimal configuration, Peierls's barrier, trigonometric polynomial.
Mathematics Subject Classification: Primary: 37J50, 37E40.
Citation: Lin Wang. Quantitative destruction of invariant circles. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021164
References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-45.   Google Scholar

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Q. Chen and C.-Q. Cheng, Regular dependence of the Peierls barriers on perturbations, J. Differential Equations, 262 (2017), 4700-4723.  doi: 10.1016/j.jde.2016.12.018.  Google Scholar

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C.-Q. Cheng and L. Wang, Destruction of Lagrangian torus in positive definite Hamiltonian systems, Geometric and Functional Analysis, 23 (2013), 848-866.  doi: 10.1007/s00039-013-0213-z.  Google Scholar

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G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.  doi: 10.1017/S0143385700007872.  Google Scholar

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M.-R. Herman, Sur la conjugation diff$\acute{e}$rentiable des diff$\acute{e}$omorphismes du cercle $\grave{a}$ des rotations, Publ. Math. IHES, 49 (1979), 5-233.  doi: 10.1007/BF02684798.  Google Scholar

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M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 144 1986.  Google Scholar

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P. H. Rabinowitz et al. NATO ASI Series C, 209. Reidel: Dordrecht, (1987), 177–202.  Google Scholar

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.  Google Scholar

[10]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[11]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, 785. Springer, Berlin, 1980. x+299 pp.  Google Scholar

[12]

L. Wang, Variational destruction of invariant circles, Discrete and Continuous Dynamical Systems-A, 32 (2012), 4429-4443.  doi: 10.3934/dcds.2012.32.4429.  Google Scholar

[13]

L. Wang, Total destruction of Lagrangian tori, Journal of Mathematical Analysis and Applications, 410 (2014), 827-836.  doi: 10.1016/j.jmaa.2013.09.018.  Google Scholar

[14]

L. Wang, Destruction of invariant circles for Gevrey area-preserving twist maps, J. Dynam. Differential Equations, 27 (2015), 283-295.  doi: 10.1007/s10884-014-9361-6.  Google Scholar

[15]

A. Zygmund, Trigonometric Series, Third Edition Volumes Ⅰ & Ⅱ combined, with a foreword by Robert Fefferman. Cambridge University Press, Cambridge, 2002.  Google Scholar

show all references

References:
[1]

V. Bangert, Mather sets for twist maps and geodesics on tori, Dynamics Reported, 1 (1988), 1-45.   Google Scholar

[2]

Q. Chen and C.-Q. Cheng, Regular dependence of the Peierls barriers on perturbations, J. Differential Equations, 262 (2017), 4700-4723.  doi: 10.1016/j.jde.2016.12.018.  Google Scholar

[3]

C.-Q. Cheng and L. Wang, Destruction of Lagrangian torus in positive definite Hamiltonian systems, Geometric and Functional Analysis, 23 (2013), 848-866.  doi: 10.1007/s00039-013-0213-z.  Google Scholar

[4]

G. Forni, Analytic destruction of invariant circles, Ergod. Th. & Dynam. Sys., 14 (1994), 267-298.  doi: 10.1017/S0143385700007872.  Google Scholar

[5]

M.-R. Herman, Sur la conjugation diff$\acute{e}$rentiable des diff$\acute{e}$omorphismes du cercle $\grave{a}$ des rotations, Publ. Math. IHES, 49 (1979), 5-233.  doi: 10.1007/BF02684798.  Google Scholar

[6]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 103-104 1983,221 pp.  Google Scholar

[7]

M.-R. Herman, Sur Les Courbes Invariantes Par Les Diff$\acute{e}$omorphismes de L'anneau, Ast$\acute{\text{e}}$risque, 144 1986.  Google Scholar

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, ed. P. H. Rabinowitz et al. NATO ASI Series C, 209. Reidel: Dordrecht, (1987), 177–202.  Google Scholar

[9]

J. N. Mather, Destruction of invariant circles, Ergod. Th. & Dynam. Sys., 8 (1988), 199-214.  doi: 10.1017/S0143385700009421.  Google Scholar

[10]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[11]

W. M. Schmidt, Diophantine Approximation, Lecture Notes in Mathematics, 785. Springer, Berlin, 1980. x+299 pp.  Google Scholar

[12]

L. Wang, Variational destruction of invariant circles, Discrete and Continuous Dynamical Systems-A, 32 (2012), 4429-4443.  doi: 10.3934/dcds.2012.32.4429.  Google Scholar

[13]

L. Wang, Total destruction of Lagrangian tori, Journal of Mathematical Analysis and Applications, 410 (2014), 827-836.  doi: 10.1016/j.jmaa.2013.09.018.  Google Scholar

[14]

L. Wang, Destruction of invariant circles for Gevrey area-preserving twist maps, J. Dynam. Differential Equations, 27 (2015), 283-295.  doi: 10.1007/s10884-014-9361-6.  Google Scholar

[15]

A. Zygmund, Trigonometric Series, Third Edition Volumes Ⅰ & Ⅱ combined, with a foreword by Robert Fefferman. Cambridge University Press, Cambridge, 2002.  Google Scholar

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