December  2012, 32(12): 4429-4443. doi: 10.3934/dcds.2012.32.4429

Variational destruction of invariant circles

1. 

Department of Mathematics, Nanjing University, No. 22 Hankou Road, Nanjing, 210093, China

Received  June 2011 Revised  November 2011 Published  August 2012

We construct a sequence of generating functions $(h_n)_{n\in\mathbb{N}}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\mathbb{N}}$ admit no invariant circles with rotation number $\omega$.
Citation: Lin Wang. Variational destruction of invariant circles. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4429-4443. doi: 10.3934/dcds.2012.32.4429
References:
[1]

V. Bangert, Mather sets for twist maps and geodesics ontori, Dynamics Reported, 1 (1988), 1-56.  Google Scholar

[2]

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

[3]

M. R. Herman, Sur la conjugation différentiable des difféomorphismes du cercle à 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 difféomorphismes de l'anneau, Astérisque, 103-104 (1983), 1-221.  Google Scholar

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M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 144 (1986), 1-248.  Google Scholar

[6]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[7]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 153-204. doi: 10.1007/BF02831625.  Google Scholar

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, 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.  Google Scholar

[10]

D. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Eletron. J., 10 (2004), Paper 3, 37 pp. (electronic).  Google Scholar

show all references

References:
[1]

V. Bangert, Mather sets for twist maps and geodesics ontori, Dynamics Reported, 1 (1988), 1-56.  Google Scholar

[2]

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

[3]

M. R. Herman, Sur la conjugation différentiable des difféomorphismes du cercle à des rotations, Publ. Math. IHES, 49 (1979), 5-233. doi: 10.1007/BF02684798.  Google Scholar

[4]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 103-104 (1983), 1-221.  Google Scholar

[5]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau, Astérisque, 144 (1986), 1-248.  Google Scholar

[6]

J. N. Mather, Existence of quasi periodic orbits for twist homeomorphisms of the annulus, Topology, 21 (1982), 457-467. doi: 10.1016/0040-9383(82)90023-4.  Google Scholar

[7]

J. N. Mather, A criterion for the non-existence of invariant circle, Publ. Math. IHES, 63 (1986), 153-204. doi: 10.1007/BF02831625.  Google Scholar

[8]

J. N. Mather, Modulus of continuity for Peierls's barrier, Periodic Solutions of Hamiltonian Systems and Related Topics, 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.  Google Scholar

[10]

D. Salamon, The Kolmogorov-Arnold-Moser theorem, Math. Phys. Eletron. J., 10 (2004), Paper 3, 37 pp. (electronic).  Google Scholar

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