Analytic expanding circle maps with explicit spectra

We show that for any $λ\in \mathbb{C}$ with $|λ|<1$ there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative powers of $λ$ and $\barλ$. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.

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