Quantitative destruction of invariant circles

<p style='text-indent:20px;'>For area-preserving twist maps on the annulus, we consider the problem on quantitative destruction of invariant circles with a given frequency <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> of an integrable system by a trigonometric polynomial of degree <inline-formula><tex-math id="M2">\begin{document}$ N $\end{document}</tex-math></inline-formula> perturbation <inline-formula><tex-math id="M3">\begin{document}$ R_N $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \|R_N\|_{C^r}&lt;\epsilon $\end{document}</tex-math></inline-formula>. We obtain a relation among <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> and the arithmetic property of <inline-formula><tex-math id="M8">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>, for which the area-preserving map admit no invariant circles with <inline-formula><tex-math id="M9">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs

We exhibit a family of convex functionals with infinitely many equal-energy $$C^1$$ C 1 stationary points that (i) occur in pairs $$v_{\pm }$$ v ± satisfying $$\det \nabla v_{\pm }=1$$ det ∇ v ± = 1 on the unit ball B in $${\mathbb {R}}^2$$ R 2 and (ii) obey the boundary condition $$v_{\pm }=\text {id}$$ v ± = id on $$ \partial B$$ ∂ B . When the parameter $$\epsilon $$ ϵ upon which the family of functionals depends exceeds $$\sqrt{2}$$ 2 , the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps $$v_{\pm }(x)$$ v ± ( x ) and prove that they are proportional to $$(\epsilon -1/\epsilon )\ln |x|$$ ( ϵ - 1 / ϵ ) ln | x | as $$x \rightarrow 0$$ x → 0 in B. The lowest-energy pairs $$v_{\pm }$$ v ± are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each $$0\le r\le 1$$ 0 ≤ r ≤ 1 , take the circle $$\{x\in B: \ |x|=r\}$$ { x ∈ B : | x | = r } to itself; a fortiori, all $$v_{\pm }$$ v ± are stationary in the class of $$W^{1,2}(B;{\mathbb {R}}^2)$$ W 1 , 2 ( B ; R 2 ) maps w obeying $$w=\text {id}$$ w = id on $$\partial B$$ ∂ B and $$\det \nabla w=1$$ det ∇ w = 1 in B.

The integrability of symplectic twist maps without conjugate points

It is proved that a symplectic twist map of the cotangent bundle $T^{\ast }\mathbb{T}^{d}$ of the $d$-dimensional torus that is without conjugate points is $C^{0}$-integrable, that is $T^{\ast }\mathbb{T}^{d}$ is foliated by a family of invariant $C^{0}$ Lagrangian graphs.