Gap invariance of a symmetric invariant lamination (Q5946887)

Let \(\mathbb{D}\) be the open unit disk in the complex plane \(\mathbb{C}\). For a subset \(A \subset \overline{\mathbb{D}}\) denote by \(\text{co} A\) the convex hull of \(A\). A subset \(S \subset \overline{\mathbb{D}}\) is called a chord if \(S = \text{co} {\{\zeta,\eta\}}\) for some \(\zeta\), \(\eta \in \partial\mathbb{D}\). If \(\zeta \neq \eta\) write \(S = \overline{\zeta\eta}\), and otherwise \(S\) is called degenerate. When \(S=\overline{\zeta\eta}\), let \(\text{ex} {S} = \{\zeta,\eta\}\). A lamination \(\mathcal{L}\) is a family of chords such that \(\bigcup \mathcal{L}\) is closed in \(\mathbb{C}\), and \(\mathcal{L}\) is non-crossing, i.e., no two elements of \(\mathcal{L}\) intersect in \(\mathbb{D}\). An element of \(\mathcal{L}\) is called a leaf, and a gap of \(\mathcal{L}\) is the closure of a component of \(\overline{\mathbb{D}} \setminus (\bigcup \mathcal{L})\). NEWLINENEWLINENEWLINEFor an integer \(d \geq 2\) let the map \(p_d : \partial\mathbb{D} \to \partial\mathbb{D}\) be defined by \(p_d(\zeta)=\zeta^d\), and for a non-degenerate chord \(S\) let \(P_dS = \text{co} {p_d(\text{ex}{S})}\). A lamination is called forward invariant under \(p_d\) if for any \(S \in \mathcal{L}\) there holds \(P_dS \in \mathcal{L}\) or \(P_dS\) is degenerate. Furthermore, \(\mathcal{L}\) is called backward invariant if for any \(S = \overline{pg} \in \mathcal{L}\) there is a collection of \(d\) disjoint chords in \(\mathcal{L}\) each joining an inverse image of \(p\) to an inverse image of \(q\). Finally, \(\mathcal{L}\) is called gap invariant if \(\operatorname {co}p_d(G \cap \partial\mathbb{D})\) is a gap of \(\mathcal{L}\), a leaf or degenerate for any gap \(G\) of \(\mathcal{L}\). NEWLINENEWLINENEWLINE\textit{C.~Bandt} and \textit{K.~Keller} [Lect. Notes Math. 1514, 1-23 (1992; Zbl 0768.58013)] have constructed a forward and backward invariant lamination under \(p_2\), and proved that it is gap invariant. In this paper, the author gives the proof of the gap invariance theorem for so-called symmetric and (forward and backward) invariant laminations under \(p_d\) for \(d \geq 2\). Furthermore, he indicates some systematic approach to study gaps of a lamination.