A system x(t) (Continuous Time) is periodic if T>0 such that x(T+t) = x(t). A system x[n] (Discrete Time) is periodic if there exists N integer>0 such that x[n+N] = x[n]
Not all complex exponentials are periodic.
Here is an example of a periodic system:
$ e^{\frac{1}{4}j*\pi*n} $ is periodic because: $ wo=(\frac{1}{4}\pi) $, $ \frac{wo}{2\pi}=\frac{1}{8} $ which is a rational number
Here is an example of a non-periodic system:
$ e^{\sqrt{3}j*\pi*n} $ is not periodic because: $ wo=\sqrt{3}\pi $ , $ \frac{wo}{2\pi} = \frac{\sqrt{3}}{2} $ which is not a rational number
