The poles of a rational discrete-time LTI system determine modes that behave like powers of the pole value.

For a pole

$$p=re^{i\omega},$$

the associated mode contains

$$p^n=r^n e^{i\omega n}.$$

The magnitude $r=|p|$ controls growth or decay. If $|p|<1$, the mode decays; if $|p|>1$, it grows.

For a causal rational discrete-time system, BIBO stability requires all poles to lie strictly inside the unit circle:

$$|p|<1.$$

Poles on the unit circle do not decay, while poles outside it generate growing modes.

Pole location therefore connects the algebra of $H(z)$ to time-domain behavior: distance from the origin controls decay or growth, while the pole angle contributes oscillation frequency.