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In this study, we partially answer the question left open in Rudin's book"Function theory in polydiscs" on the structure of invariant subspaces of theHardy space $H^2(U^n)$ on the polydisc $U^n$. We completely describe allinvariant subspaces generated by a single function in the polydisc. Then, usingour results, we give the unitary equivalence of this type of invariant subspaceand a characterization of outer functions in $H^2(U^n)$.

Invariant subspaces generated by a single function in the polydisc