Symmetric band complexes of thin type and chaotic sections which are not quite chaotic (Q2788928)

A band is a rectangle \(\mathcal B=[a,b]\times[0,1]\subset\mathbb R^2\) (\(a\leq b\)), equipped with the \(1\)-form \(dx\), where \(x\) is the first coordinate in \(\mathbb R^2\). The horizontal sides \([a,b]\times\{0\}\) and \([a,b]\times\{1\}\) are called bases. A band complex is a \(2\)-complex \(X\) obtained by gluing a family of disjoint bands \(\mathcal B_i=[a_i,b_i]\times[0,1]\) to a union \(D\) of disjoint closed intervals (the support multi-interval), so that the base of every \(\mathcal B_i\) is isometrically glued to a subinterval of \(D\), preserving the orientation. Thus \(X\) can be equipped with the \(1\)-form defined by \(dx\) on the bands and \(D\). We also get a singular foliation \(\mathcal F_X\) whose leaves are the maximal connected components where the restriction of \(dx\) vanishes. Its singular set, \(\text{sing }X\), consists of the points where \(\mathcal F_X\) is not locally given by a fibration over an open interval; i.e., the set given by the vertical sides of the bands. If the leaves are simply connected, then \(X\) is called annulus free. The rank of \(X\) is the dimension of the \(\mathbb Q\)-vector space consisting of the integrals \(\int_cdx\) for all \(c\in H_1(X,\text{sing }X;\mathbb Z)\). An isomorphism of band complexes has the obvious definition. It is said that \(X\) is symmetric if there is an involution \(\tau:X\to X\) that takes bands to bands and satisfies \(\tau^*dx=-dx\). A free arc \(J\subset D\) is a maximal open interval that is covered by only one of the bases of bands, and all other bases are disjoint from \(J\). In this case, by removing the part of the band over \(J\), we get a new band complex \(X'\), which is said to be obtained from \(X\) by a collapse from a free arc. An annulus free band complex \(X\) is said to be of thin type if every leaf of the foliation \(\mathcal F_X\) is everywhere dense in \(X\), and there is an infinite sequence \(X_0=X,X_1,X_2,\dots\) in which every \(X_i\) is a band complex obtained from \(X_{i-1}\) by a collapse from a free arc (\(i \geq1\)).NEWLINENEWLINEThe main theorem states that there exist uncountably many symmetric band complexes \(X\) such that \(X\) consists of \(3\) bands, \(X\) has rank \(3\), \(X\) is of thin type, and almost any leaf of \(\mathcal F_X\) is a two-ended tree. This is applied to construct a \(3\)-periodic surface in \(\mathbb R^3\) with a plane direction such that the surface has a central symmetry, and the plane sections of the chosen direction are chaotic and consist of infinitely many connected components. Moreover the typical connected components of the sections have an asymptotic direction, which is due to the fact that the corresponding foliation on the surface in the \(3\)-torus is not uniquely ergodic.