EVENT DETAILS AND ABSTRACT

Title: A Z^2 generalization of the Morse-Hedlund theorem
Speaker: Van Cyr
Speaker Info: Northwestern University
Brief Description:
Special Note:
Abstract:

If $\A$ is a finite alphabet and $f:\Z\to\A$, the {\em block complexity function} of $f$ is the function $P_f: \to $ defined by $$ P_f(n)=\#\text{(distinct words of length $n$ occurring in $f$)}. $$ A classical theorem of (one-dimensional) symbolic dynamics is the Morse-Hedlund theorem which says that $f$ is periodic if and only if there exists $n_0\in $ so that $P_f(n_0)\leq n_0$. A 1997 conjecture of M. Nivat says that a similar phenomenon happens in $\Z^2$-symbolic dynamical systems: for $f:\ZZ\to\A$ and $$ P_f(n,k):=\#\text{(distinct $n\times k$-rectangular words occurring in $f$)}, $$ if there exist $n_0, k_0\in $ so that $P_f(n_0, k_0)\leq n_0k_0$ then $f$ is periodic in some direction. Many partial results are known but this conjecture has proven difficult to show in general.

In this talk I will discuss some recent work (joint with B. Kra) in which we prove the following generalization of the Morse-Hedlund theorem: for $f:\ZZ\to\A$, $f$ is doubly periodic (i.e. has finite $\ZZ$-orbit) if and only if $f$ has no persistent lines (a definition which will be discussed) and there exist $n_0, k_0\in $ so that $P_f(n_0, k_0)\leq n_0k_0$. The main application of this theorem is toward a new attack on Nivat's conjecture.