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Register a new userDestruction of invariant circles for Gevrey area-preserving twist maps
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Lin Wang
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In this paper, we show that for exact area-preserving twist maps on annulus, the invariant circles with a given rotation number can be destroyed by arbitrarily small Gevrey-$alpha$ perturbations of the integrable generating function in the $C^r$ topology with $r<4-frac{2}{alpha}$, where $alpha>1$.
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We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.
Poincares last geometric theorem (Poincare-Birkhoff Theorem) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under $f$ at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincares geometric theorem, our result also has some applications to reversible systems.
We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $cin H^1(mathbb T,mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(sigma)in H^1(mathbb T,mathbb R)$ with $c(sigma)$ continuous and monotonic and [ partial_tu_c+H(x,partial_x u_c+c,t)=alpha(c),quad text{a.e. } (x,t)inmathbb T^2, ] such that sequence of weak KAM solutions ${u_c}_{cin H^1(mathbb T,mathbb R)}$ is $1/2-$Holder continuity of parameter $sigmain mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving [ left{ begin{aligned} &dot{x}(s)in text{co} Big[partial_pHBig(x(s),c+D^+u_cbig(x(s),s+tbig),s+tBig)Big], & &x(0)=x_0,quad (x_0,t)inmathbb T^2,& end{aligned} right. ] we evaluate it by a uniquely identified rotational number $omega(c)in H_1(mathbb T,mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
We consider the minimal average action (Mathers $beta$ function) for area preserving twist maps of the annulus. The regularity properties of this function share interesting relations with the dynamics of the system. We prove that the $beta$-function associated to a standard-like twist map admits a unique $C^1$-holomorphic complex extension, which coincides with this function on the set of real diophantine frequencies.
The stickiness effect suffered by chaotic orbits diffusing in the phase space of a dynamical system is studied in this paper. Previous works have shown that the hyperbolic structures in the phase space play an essential role in causing the stickiness effect. We present in this paper the relationship between the stickiness effect and the geometric property of hyperbolic structures. Using a two-dimensional area-preserving twist mapping as the model, we develop the numerical algorithms for computing the positions of the hyperbolic periodic orbits and for calculating the angle between the stable and unstable manifolds of the hyperbolic periodic orbit. We show how the stickiness effect and the orbital diffusion speed are related to the angle.
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