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We present results concerning the cyclic behaviour of weighted shifts and composition operators on the disc algebra. Most of our proofs use the representations of the dual space of the disc algebra as Banach spaces of analytic functions on the unit disc. Taking into account that no composition operator on the disc algebra can be hypercyclic, the results we obtain show that the composition operators induced by linear fractional maps with an interior fixed points have the same cyclic behaviour on the disc algebra as they do on the Hardy space, as described in Bourdon and Shapiro [Cyclic phenomena for composition operators. Mem Am Math Soc. 1997;125(596)], while the operators induced by linear fractional maps with boundary fixed points have the same cyclic behaviour on the disc algebra as they do on the Dirichlet space, as described in Gallardo-Gutiérrez and Montes-Rodríguez [The role of the spectrum in the cyclic behavior of composition operators. Mem Am Math Soc. 2004;167(791)].

Cyclic behaviour of certain operators on the disc algebra