Iterations of Piecewise Maps

I will present a few aspects of the reachability problem for piecewise affine maps. These are functions f from the unit interval to itself, such that there is a partition of the domain into intervals where f is affine, i.e. of the form ax+b. They are related to hybrid systems, expansions of numbers into non-integral bases, discount-sum automata etc. The reachability problem asks for a procedure which inputs such a map, a source and target point, and decides whether by iterating the map on the source we reach the target. No such procedure is known. I will give one for injective maps.
The talk is based on joint work with Faraz Ghraremani and Joël Ouaknine, published in LICS 2023.