Space-filling curves and geodesic laminations. II: Symmetries (Q431188)

Let \((X; \psi_1, \ldots, \psi_k)\) be an iterated function system (IFS) consisting of \(k\) contractions and let \(\mathfrak{R}\) be its attractor. The IFS possesses the \textit{common point property} if {\parindent=6mm \begin{itemize} \item{} \(X:= \mathbb{R}^d\), \(2\leq d\in \mathbb N\); \item {} the mappings \(\psi_i\) are affine; \item {} the IFS satisfies the open set condition; \item {} \(\bigcap_{i=1}^k \psi_i (\mathfrak{R}) = \{z\}\) and there exists one point \(y\in \mathfrak{R}\) such that \(\psi_1 (y) = \cdots = \psi_k (y) = z\). \end{itemize}} In a previous publication [\textit{V. F. Sirvent}, Geom. Dedicata 135, 1--14 (2008; Zbl 1147.28007)], the author studied IFSs with the common point property and showed that they admit geodesic laminations of the disc. In this follow-up paper, the relationship between the symmetries of the laminations and the underlying fractal, i.e., \(\mathfrak{R}\), is investigated. It is proved that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of \(\mathfrak{R}\). The results are extended to a larger class of fractal attractors employing the concept of a sub-IFS.