Abstract: In this talk I will discuss my previous and current work with Chris Mouron on topological entropy of certain maps on inverse limits and dynamical properties of commuting maps. If a map on an inverse limit is represented by a commutative "straight-down" diagram, it is easy to compute its topological entropy (Ye 1995), but there are no such simple tools in general. We study a class of inverse limit maps we call "diagonal maps", and show that in a special case when a map is given by a "strongly commuting" diagonal diagram, the topological entropy can be computed as the limit of entropies of its straight-down components, which are in this case set-valued functions. We will analyze in more detail strong commutativity when all factor spaces are intervals. In particular, we show that piecewise monotone strongly commuting maps on the interval have a common fixed point. There exist commuting maps on the interval without common fixed points (Boyce and Huneke 1967), but we still do not know if there are such maps without common periodic points. There are actually many open questions about commutativity of maps (even on simple spaces, such as intervals). We will discuss some of those open questions and their implications.