Trace formula for periodic points of a subshift ($\#\mathrm{Fix}({\sigma }^n)=\mathrm{tr}(A^n)$)

A fixed point of the $n$th iteration of a map of a subshift of finite type is determined uniquely by its length-$n$ block with a permitted closing transition. Summing the counts of length-$n$ closed walks over their starting symbols gives the displayed trace. This counts points whose least period divides $n$. If $P_n$ counts points of least period exactly $n$, then $\mathrm{tr}(A^n)={\sum }_{d\mid n}{P}_d$, and Möbius inversion gives $P_n={\sum }_{d\mid n}{\mu }_{\mathrm{M}}(n/d)\mathrm{tr}(A^d)$. The number of distinct least-period-$n$ periodic orbits is $P_n/n$.