In symbolic dynamics, one considers a finite set of symbols and the shift map on a space of (bi-)infinite sequences of those symbols (e.g. bi-infinite sequences of 0's and 1's). The shift map restricted to a closed subspace is called a subshift. General dynamical systems can be modeled by subshifts where the symbols represent various states of the system and the shift maps represents the passage of one unit of time.
The set of finite strings of symbols, or words, that appear in a given subshift can give us insight into its dynamics. This talk will focus on the complexity of a subshift, reflected by the sequence p(n) of number of words of length n appearing in the subshift. We will focus on situations where bounds on the limsup of p(n)/n imply restrictions on the types of dynamics we can see within the subshift. With differing hypotheses various authors have considered this question as it relates to minimal components, ergodic invariant measures, dynamical spectrum, and other properties. I will present a survey of some of these results with a particular emphasis on results where limsup p(n)/n=3/2 is a threshold for dynamical properties.
*The Math Colloquium will be available live via Zoom at https://uiowa.zoom.us/j/99206823329 and recordings of the sessions will be available at https://uicapture.hosted.panopto.com/Panopto/Pages/Sessions/List.aspx?folderID=b23b61cc-a48a-4de7-99fc-ae46015b8b6e usually within a couple hours of the colloquium finishing.