Output of discrete-time LTI system guaranteed to be same form as input? I know that in the continuous-time context, if I supply a complex exponential input to a Linear Time Invariant system, the output will be of the same form as the input - for example, if the input is $x(t)=Ae^{j \omega_0 t+\phi}$, the output will be of the form $y(t)=|H(j\omega_o)|\centerdot A \centerdot e^{j(H(s)\omega_0 t+\phi +\theta_H)}$ where $H(s)|_{s=j\omega_0} = |H(j\omega_0)|e^{j\theta_H}$ is the transfer function of the system evaluated at the frequency $j\omega_0$. My point is that the system does not modify the input frequency.

Does the same hold true in the discrete-time context? If I supply an input $x[n]=a^n$ to an LTI system, will the output also contain $a^n$? Or is it possible to supply $a^n$ and get the output $b^n +c^n$ where $a\ne b\ne c$ (for example, $x[n]=(1/6)^n$, $y[n]=B(1/2)^n + C(1/3)^n$)?

Edit: I've posted a related question that contains the issue that actually prompted this question...
The point is that a complex exponential function is an eigenfunction of an LTI system link. Therefore the frequency is not changing, i.e., the LTI system translates to a (complex-valued) gain $H(j\omega_0)$.

For discrete systems, the eigenfunctions are sampled exponential functions link, i.e., 
$x[n] = A\cdot e^{j\omega_n n}$, where $\omega_0=2\pi f/f_s$ is a normalized frequency. The output of a discrete LTI system is then $y[n]=H(e^{j\omega_n})\cdot A \cdot e^{j\omega_nn}$.

EDIT:
A discrete input signal x[n] can be written as a weighted sum of complex exponentials, i.e., $X(e^{j\omega}) = \sum_{n}x[n]e^{j\omega}$ link. This transformation is in fact a change of basis. Driving such a signal thru an LTI system will weight each exponential function with the complex-valued gain $H(e^{j\omega})$. Thus, no additional frequencies will be introduced.

Note (a bit out of scope, but good to realize): 
When dealing with sampling a contuous-time signal and dealing with finite length signals, then you may observe additional frequencies due to aliasing and windowing.