Symmetries in \(k\)-bonacci adic systems (Q2855532)

In this paper the author studies some questions of symmetry for the so called \(k\)-bonacci adic systems \(({\mathcal R}_{k},T_{k})\) (for \(k \in {\mathbb N}, k \geq 3)\), and investigates how the so obtained results apply to the geometric realizations of \(k\)-bonacci adic systems, the so called Rauzy fractals. Note that the space \({\mathcal R}_{k}\) is the symbolic dynamical systems with symbols in the alphabet \(\{0,1\}\), consisting of all one-sided infinite sequences of zeros and ones such that there are no \(k\) consecutive ones, and the map \(T_{k}\) represents ``addition by \(1\)''. Let \({\mathcal P}_{k}\) denote the maximal subsystem invariant under the involution \(\psi: \{0,1\}^{{\mathbb N}} \to \{0,1\}^{{\mathbb N}}\), given by \(\psi(0):=1\) and \(\psi(1):=0\). The main result of the paper is to show the existence of a continuous map \(S_{k}\) on \({\mathcal P}_{k}\) such that the dynamical system \(({\mathcal P}_{k},S_{k})\) is semi-conjugated to the system \(({\mathcal R}_{k-1}, T_{k-1})\) and such that the semi-conjugacy is a \(2-1\) map. The paper finishes by giving some consequences of this result for Rauzy fractals associated with \(k\)-bonacci adic systems.