Heteroclinic orbits and chaotic invariant sets for monotone twist maps
Tifei Qian Zhihong Xia
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 69-95 doi: 10.3934/dcds.2003.9.69
We consider the monotone twist map $\bar f$ on $(\mathbb R/\mathbb Z)\times R$, itslift $f$ on $R^2$ and its associated variational principle $h:\mathbb R^2\to\mathbb R$ through its generating function. By working with the variationalprinciple $h$, we first show that for an adjacent minimal chain$\{(u^k, v^k)\}_{k=s}^t$ of fixed points of $f$, if there exists abarrier $B_k$ for each adjacent minimal pair $u^k < u^{k+1}$, $ s \le k \le {t-1} $, then there exists a heteroclinic orbit between $(u^s, v^s)$ and$(u^t, v^t)$, then by assuming that there is a barrier for any twoneighboring globally minimal critical points and $m$ is sufficientlylarge, we construct an invariant set $\Lambda^m\subset (\mathbb R/\mathbb Z)\times\mathbb R$ such that the shift map of the $n$-symbol space is a factor of$\bar f^m|_{\Lambda^m}$, where $n$ is the total number of the globallyminimal fixed points of $\bar f$.
keywords: chaotic invariant sets. Hamiltonian dynamics variational method Twist maps

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