Analytic expanding circle maps with explicit spectra

Abstract

Abstract. We show that for any λ ∈ 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 λ. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.

From dynamical systems with time-varying delay to circle maps and Koopman operators

“…The spectrum of a Koopman operator depends on the dynamics of the underlying map and can be computed analytically, for example, for maps with a unique (attractive) fixed point, as mentioned before, and for purely expanding circle maps [35]. In our case of invertible access maps there are only two possible types of dynamics, which determine our delay types, conservative and dissipative delay.…”

Spectral structure of transfer operators for expanding circle maps

“…As we shall see presently, the above choices of the annuli guarantee that L is a well-defined linear operator which maps H 2 (A) compactly to itself. In order to show this, we shall employ a factorization argument, similar to the ones used in [4,21]. Let H ∞ (A ′ ) be the Banach space of bounded holomorphic functions on A ′ equipped with the supremum norm.…”

“…Example 5.7. For the family of maps B(z) = z(µ − z)/(1 − µz) considered in [21], the restriction B| T is an expanding circle map for any µ ∈ D. The attracting fixed points are z 0 = 0 andẑ 0 = ∞ with λ(z 0 ) = µ and λ(ẑ 0 ) = µ. Thus σ(L) = {µ n : n ∈ N 0 } ∪ {µ n : n ∈ N} ∪ {0} .…”

“…preserves and acts compactly on certain spaces of holomorphic functions [17,18,21]. The spectrum of L can be understood by passing to its (Banach space) adjoint operator L * .…”

Generic non-trivial resonances for Anosov diffeomorphisms

“…More recently, Bandtlow et. al [2,20] calculated the resonances of real analytic expanding maps T : S → S on the unit circle S explicitly for Blaschke products. Their transfer operator acts on the Hardy space of holomorphic functions on the annulus.…”