Noncommutative dynamical systems with two generators and their applications in analysis (Q1431406)

The author studies the dynamics defined by two noncommuting interval maps \(\delta_1\) and \(\delta_2\). In general, each point has two possible images and the number of orbits of a given length grows exponentially with the length. The dynamics can be restricted by introducing two disjoint bounded guiding sets \(\mathcal{T}_1\) and \(\mathcal{T}_2\). An orbit then is called {proper} if the iterate \(t_{k+1}\) is \(t_{k+1}=\delta_j(t_k)\) whenever \(t_k\in\mathcal{T}_j\). The main theorem gives necessary and sufficient conditions under which for arbitrary initial conditions there exists a proper orbit converging to the boundary of the interval. In a last part, it is shown how problems from functional equations and from integral geometry can be reformulated under appropriate conditions such that they fit in the dynamical framework discussed before.