Heteroclinic orbits and chaotic invariant sets for monotone twist maps (Q1865946)

The authors consider monotone twist maps as defined by \textit{J. N. Mather} in [J. Am. Math. Soc. 4, 207-263 (1991; Zbl 0737.58029)], that is, \(C^{1}\) exact area-preserving, orientation-preserving, positive monotone twist diffeomorphisms of the infinite cylinder that preserve the ends, twist each end infinitely, and have a uniform lower bound for the amount of twisting. Given a monotone twist map \(\widehat{f}\) on \((\mathbb{R}/\mathbb{Z})\times \mathbb{R}\), by working with its lift \(f\) and its associated variational principle \(h\), they prove the following results: (1) For an adjacent minimal chain \(\{(u^{k},v^{k})\}_{k=s}^{t}\) of fixed points of \(f\), if there exists a barrier \(B_{k}\) for each adjacent minimal pair \(u^{k}<u^{k+1}\), \(s\leq k \leq t-1\), then there exists a heteroclinic orbit between \((u^{s},v^{s})\) and \((u^{t},v^{t})\). (2) Suppose that \(\{(u^{k},v^{k})\}_{k=0}^{n-1}\) enumerates all the globally minimal fixed points of \(f\) such that \(0\leq u^{0}< \cdots < u^{k}< u^{k+1}< \cdots < u^{n-1}<1\). If there exists a barrier for each adjacent minimal pair \(u^{k}<u^{k+1}\), \(0\leq k\leq n-2\), then there is \(N>0\) such that for \(m\geq nN\), there exists an invariant set \(\Lambda^{m}\subset (\mathbb{R}/\mathbb{Z})\times \mathbb{R}\) such that the shift map on the \(n\)-symbol space \(\Sigma^{n}\) is a factor of \(\widehat{f}^{m}|_{\Lambda^{m}}\).