On the Orbits of Multiplicative Pairs

We characterize all pairs of completely multiplicative functions $f,g:\mathbb{N}\to\mathbb{T}$ such that the orbit closure $\overline{\{(f(n),g(n+1))\}_{n\ge 1}} \neq \mathbb{T}\times \mathbb{T}.$ In so doing, we settle an old conjecture of Zoltán Daróczy and Imre Kátai.

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