# Analysis, Geometry, and Dynamical Systems Seminar

### Inﬁnite alphabet ultragraph edge shift spaces: relations to $C^\ast$-algebras and chaos

We explain the notion of ultragraphs, which generalize directed graphs, and use this combinatorial object to deﬁne a notion of (one-sided) edge shift spaces (which, in the ﬁnite case, coincides with the edge shift space of a graph). We then go on to show that these shift spaces have some nice properties, as for example metrizability and basis of compact open sets. We examine shift morphisms between these shift spaces: we give an idea how to show that if two (possibly inﬁnite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph $C^\ast$-algebras are isomorphic. Finally we describe Li-Yorke chaoticity associated to these shifts and remark that the results obtained mimic the results for shifts of finite type over finite alphabets (what is not the case for infinite alphabet shift spaces with the product topology).