 Solution

A return path with $u$ upward steps and $v$ downward steps satisfies $au=bv$. Since $a,b$ are coprime, this is equivalent to $u=bk$ and $v=ak$ for an integer $k\ge 1$. Its length is therefore $k(a+b)$.

Conversely, for each such $k$, any ordering of $bk$ upward and $ak$ downward steps is an allowed positive-probability return path. The possible strictly positive return times are exactly the positive multiples of $a+b$. By the definition of the period of a state in a Markov chain as their greatest common divisor,In particular, none of the states is aperiodic.