There is a rich history of studying dynamical systems through the lens of operator algebras, and particularly through C*-algebras. For instance, in the work of Giordano, Matui, Putnam and Skau, C*-algebras were used as a key tool for classifying Cantor minimal $\mathbb{Z}^d$ systems up to various notions of orbit equivalence. Another successful study was conducted by Cuntz and Krieger, where subshifts of finite type (SFTs) are interpreted through C*-algebras of directed graphs, and invariants studied in symbolic dynamics naturally arise from these C*-algebras.

In this talk we will explain how Cuntz--Krieger algebras are used to encode directed graphs, as well as dynamical properties of SFTs, by describing a hierarchy of classification results for these C*-algebras with appropriate added structure.

After this is done, we will discuss an ongoing project where we associate operator algebras to random walks, similarly to how Cuntz-Krieger algebras are constructed from directed graphs. It turns out that these algebras are intimately related to the limiting behavior of ratios of transition probabilities of the random walk, a behavior studied by many authors in probability over the years. We will see how this limiting behavior is related to the structure and classification of these algebras, and how this study is yet another example for providing new insight, this time in the theory of random walks.