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Hypercyclic Toeplitz operators

We study hypercyclicity of the Toeplitz operators in the Hardy space $H^2(\mathbb{D})$ with symbols of the form $p(\bar{z}) +ϕ(z)$, where $p$ is a polynomial and $ϕ\in H^\infty(\mathbb{D})$. We find both necessary and sufficient conditions for hypercyclicity which almost coincide in the case when ${\rm deg}\, p =1$.