Non-leading eigenvalues of the Perron-Frobenius operators for beta-maps

TL;DR

This paper studies the non-leading eigenvalues of Perron-Frobenius operators for beta-maps, showing their prevalence, continuity properties, and explicit eigenfunction formulas, revealing new spectral characteristics.

Contribution

It demonstrates the density of beta values with non-leading eigenvalues, proves their Hölder continuity, and provides explicit formulas for eigenfunction evaluations, advancing spectral analysis of beta-maps.

Findings

Non-leading eigenvalues are common for beta-maps.

Eigenvalues vary Hölder continuously with beta, but are non-differentiable.

Eigenfunctionals for non-leading eigenvalues cannot be represented by complex measures.

Abstract

We consider the Perron-Frobenius operator defined on the space of functions of bounded variation for the beta-map $\tau_\beta(x)=\beta x$ (mod $1$), for $\beta\in(1,\infty)$, and investigate its isolated eigenvalues except $1$, called non-leading eigenvalues in this paper. We show that the set of $\beta$'s such that the corresponding Perron-Frobenius operator has at least one non-leading eigenvalue is open and dense in $(1,\infty)$. Furthermore, we establish the H\"older continuity of each non-leading eigenvalue as a function of $\beta$ and show in particular that it is continuous but non-differentiable, whose analogue was conjectured by Flatto et.al. in \cite{Fl-La-Po}. In addition, for an eigenfunctional of the Perron-Frobenius operator corresponding to an isolated eigenvalue, we give an explicit formula for the value of the functional applied to the indicator function of every interval. As its application, we provide three results related to non-leading eigenvalues, one of which states that an eigenfunctional corresponding to a non-leading eigenvalue can not be expressed by any complex measure on the interval, which is contrast to the case of the leading eigenvalue $1$.