There are various ways in which one can describe the
simplicity/complexity of a dynamical system. One is in terms of qualitative
dynamical properties, such as periodicity/equicontinuity (simple) or
mixing/independence (complex). For symbolically defined dynamical systems
called subshifts, there is a more quantitative measure called the word
complexity function, which counts the number of finite words appearing in
the system as a function of length. Naturally, there are relations between
these notions, but the exact nature of some is quite surprising and
unexpected. I’ll cover some of these relationships, including the classical
Morse-Hedlund theorem and more recent work of myself and Creutz. No prior
knowledge is required.