An elementary counterexample to the finiteness conjecture
Volume: 24 • Number: 4 • Pages: 963-970
We prove that there exist ( infinitely many) values of the real parameters a and b for which the matrices [GRAPHICS] 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.