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We show that for any $\lambda \in \mathbb{C}$ with $|\lambda|<1$ there existsan analytic expanding circle map such that the eigenvalues of the associatedtransfer operator (acting on holomorphic functions) are precisely thenonnegative powers of $\lambda$ and $\bar{\lambda}$. As a consequence we obtaina counterexample to a variant of a conjecture of Mayer on the reality ofspectra of transfer operators.

Analytic expanding circle maps with explicit spectra