Dynamical Systems Seminar

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.