We look at constructions of aperiodic subshifts of finite type (SFTs) on fundamental groups of graph of groups. In particular, we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by Whyte and on two constructions of strongly aperiodic SFTs on $\mathbb {F}_n\times \mathbb {Z}$ and $BS(m,n)$ of our own. Our two constructions rely on a path-folding technique that lifts an SFT on $\mathbb {Z}^2$ inside an SFT on $\mathbb {F}_n\times \mathbb {Z}$ or an SFT on the hyperbolic plane inside an SFT on $BS(m,n)$. In the case of $\mathbb {F}_n\times \mathbb {Z}$, the path folding technique also preserves minimality, so that we get minimal strongly aperiodic SFTs on unimodular GBS groups.

Multiwords are words in which a single symbol can be replaced by a nonempty set of symbols. They extend the notion of partial words. A word w is certain in a multiword M if it occurs in every word that can be obtained by selecting one single symbol among the symbols provided in each position of M. Motivated by a problem on incomplete databases, we investigate a variant of the pattern matching problem which is to decide whether a word w is certain in a multiword M. We study the language CERTAIN(w) of multiwords in which w is certain. We show that this regular language is aperiodic for three large families of words. We also show its aperiodicity in the case of partial words over an alphabet with at least three symbols.

We develop a notion of Morita equivalence for general C$^{\ast}$-correspondencesover C$^{\ast}$-algebras. We show that if twocorrespondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, theToeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz--Pimsner algebras. Conversely, if the tensor algebras arestrongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C$^{\ast}$-algebras, then the correspondences are Morita equivalent. Thisgeneralizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure-preserving transformations. The notion ofaperiodicity, which also generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated. 1991 Mathematics Subject Classification: 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij of the total amount of flow which arrived at node i during the previous time period. The fraction pij whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij are discussed.