| Adic transformations (or simply adics) are dynamical systems defined on the space of infinite paths in a graph called a Bratteli diagram. We review the dynamical properties of the Pascal adic, which is an example of a nonstationary adic where the number of vertices increases from one level to the next. We describe this system in terms of its Bratteli diagram and by cutting and stacking, and we give a new proof that the nonatomic invariant ergodic probability measures are the Bernoulli measures. We also establish a result on the asymptotics of return times into typical cylinders.;Then we introduce a class of nonstationary adics called the generalized Pascal adic transformations. For these systems we show that the set of nonatomic invariant ergodic probability measures is formed by a certain one-parameter family of Bernoulli measures, generalizing the previous result on the Pascal adic.;We show that the Pascal adic is isomorphic to a symbolic system with the shift transformation and defined by countably many substitutions. For this countable-substitution subshift we show that the complexity function is asymptotic to a cubic. Next, we show that the frequency of appearances of a given block in the language satisfies a property stronger than ergodicity that we call local unique ergodicity. Finally, we prove that, as a topological dynamical system (which is not quite minimal), the Pascal adic is topologically weakly mixing. |
