I'm working out of Devaney's Introduction to Chaotic Systems, and one of the problems I'm working on is to construct a subshift of finite type in $\Sigma_3$ with no fixed or period two points, but with points of period 3.
I don't know how to approach this problem, but it seems to me that if the number of periodic points of a subshift is given by the trace of its transition matrix, then this problem has no solution, since it's asking me to find a matrix $A$ such that $trace$(A)$=$$trace$($A^2$)$=0$, but have $trace(A^3)\neq\=0$, but this is impossible for a $3x3$ matrix.
What am I misunderstanding here? Thanks.
Echoing Siming Tu's comment. Such a matrix exists. Let $A$ be a square matrix with $a_{1,2}=a_{2,3}=a_{3,1} = 1,$ else 0. This matrix shifts the top row to the bottom under matrix multiplication. So that the trace of $A$ and $A^2$ are $0$ as everything is off the diagonal. However, $A^3$ is the identity, so it has trace $1$.