Analytic expanding circle maps with explicit spectra

TL;DR

This paper constructs explicit analytic expanding circle maps with transfer operators whose spectra are precisely the nonnegative powers of a given complex number, providing a counterexample to a conjecture about the reality of transfer operator spectra.

Contribution

It introduces a method to explicitly construct expanding circle maps with prescribed spectral properties, challenging existing conjectures.

Findings

Eigenvalues are exactly the nonnegative powers of any given complex number with magnitude less than 1.

Provides a counterexample to a conjecture on the reality of transfer operator spectra.

Demonstrates the existence of explicit maps with tailored spectral characteristics.

Abstract

We show that for any $\lambda \in \mathbb{C}$ with $|\lambda|<1$ 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 $\lambda$ and $\bar{\lambda}$. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.