Adic dynamics on the Catalan graph

In this paper, we study the dynamics of the adic map on the Catalan graph. The Catalan graph is formed by taking the right half of the well-studied Pascal graph, and the Catalan numbers arise as path counts. For the adic map on the Catalan graph, the we show that the edge weights for invariant measures depend not only on the level, but also the index of the source and range vertices. This is in contrast to the Pascal adic systems and many other examples that have been explored. We prove a number of results about the dynamics and complexity of the associated measure-theoretic dynamical systems.