ON EXAMPLES OF RANK-TWO SYMBOLIC SHIFTS JAMES LENG, CESAR E. SILVA, AND YUXIN WU Abstract. We study rank-two symbolic systems (as topological dynamical systems) and prove that the Thue-Morse sequence and quadratic Sturmian sequences are rank-two and define rank-two sym- bolic systems. 1. Introduction Rank-one measure-preserving transformations have played an important role in ergodic theory since the pioneering work of Chacón [4]. In this paper, rather than considering the notion of rank- one in measurable dynamics we are interested in studying rank-one and higher rank systems strictly from the point of view of topological dynamics, in particular as symbolic shifts. Of course, sym- bolic systems have been used extensively in ergodic theory. In [13], Kalikow discusses a symbolic model for rank-one measure-preserving transformations, and Ferenczi [8] in his survey on rank-one finite measure-preserving transformations mentions the symbolic definition for rank-one transfor- mations. Later in [3], Bourgain used a class of symbolic rank-one transformations for which he proved the Moebius disjointness law, and rank-one symbolic shifts are also considered in [1, 5, 7]. It was in [11] that Gao and Hill started a systematic study of (non-degenerate) rank-one shifts as topological dynamical systems, and proved several properties for rank-one symbolic shifts. In this paper we study higher rank systems and prove that the system defined by the Thue-Morse sequence and systems defined by quadratic Sturmian sequences are rank-two. The terminology “rank-one" comes from “rank-one" cutting and stacking systems [4]. As shown by Kalikow [13], one can encode cutting and stacking systems as a shift on a symbolic system and he shows that the two systems are measurably isomorphic when the symbolic sequence is non-periodic. Gao and Hill introduced the notion of (symbolic) rank-one shifts as an extension to earlier ideas and developed various results associated to symbolic rank-one systems. We start by first defining rank-one words using the definition provided by Gao and Hill in [11]. Definition 1.1. Let F be the set consisting of all finite words in the alphabet {0, 1} that begin and end with 0. Let V ∈{0, 1} N . We say that V is built from v ∈F if there exists a sequence {a i } i≥1 of natural numbers such that V = v1 a 1 v1 a 2 v ··· . Let A V = {v ∈F : V is built from v}. We say that the infinite word V is rank-one if A V is infinite. Example. The infinite word U = 01010101010 ··· Date: August 2, 2021. 2010 Mathematics Subject Classification. Primary 37A40; Secondary 37A05, 37A50. Key words and phrases. Symbolic systems, rank-one, Thue-Morse. 1 arXiv:2010.05165v1 [math.DS] 11 Oct 2020