For a finite alphabet \({{\bf A}}\) and \(ta\colon {\bf Z} →
{{\bf A}}\), the Morse-Hedlund Theorem states that \(ta\) is periodic if
and only if there exists \(n\in{{\bf N}}\) such that the block complexity
function \(P_ta(n)\) satisfies \(P_ta(n)\leq n\). In dimension two, a
conjecture of M. Nivat states that if there exist \(n,k\in{{\bf N}}\) such
that the \(n × k\) rectangular complexity function,
\(P_{ta}(n,k)\), satisfies \(P_{ta}(n,k)\leq nk\), then \(ta\) is
periodic.
There have been a number of attempts to prove Nivat's conjecture over the
past 15 years, but the problem has proven difficult. In this talk I will
discuss recent joint work with B. Kra in which we associate a \({{\mathbf
Z}^2}\)-dynamical system with \(ta\) and show that if there exist
\(n,k\in{{\bf N}}\) such that \(P_{ta}(n,k)\leq nk\), then the periodicity
of \(ta\) is equivalent to a statement about the expansive subspaces of
this action. The main result is a weak form of Nivat's conjecture: if there
exist \(n,k\in{{\bf N}}\) such that \(P_{ta}(n,k)\leq rac{nk}{2}\), then
\(ta\) is periodic.
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