An Elementary Counterexample to the Finiteness Conjecture
Vincent D. Blondel, Jacques Theys, Alexander Vladimirov · SIAM Journal on Matrix Analysis and Applications · 2003
We prove that there exist (infinitely many) values of the real parameters a and b for which the matrices ( \begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array} \right) \qquad \mbox{ and } \qquad b \left( \begin{array}{ll} 1 & 0 \\ 1 & 1 \end{array} \right) $$ have the following property: all infinite periodic products of the two matrices converge to zero, but there exists a nonperiodic product that doesn't. Our proof is self-contained and fairly elementary; it uses only elementary facts from the theory of formal languages and from linear algebra. It is not constructive in that we do not exhibit any explicit values of a and b with the stated property; the problem of finding explicit matrices with this property remains open.