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

Part of: Dynamical systems with hyperbolic behavior Ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  29 August 2025

SHINTARO SUZUKI Open the ORCID record for SHINTARO SUZUKI [Opens in a new window]
SHINTARO SUZUKI*
Affiliation:
Department of Mathematics, https://ror.org/00khh5r84 Tokyo Gakugei University, 4-1-1 Nukuikitamachi Koganeishi, Tokyo 184-8501, Japan
*
e-mail: shin05@u-gakugei.ac.jp
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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 $ 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 Hölder 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, Lagarias and Poonen in [The zeta function of the beta transformation. Ergod. Th. & Dynam. Sys. 14 (1994), 237–266]. 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 cannot be expressed by any complex measure on the interval, which is in contrast to the case of the leading eigenvalue $1$.

Keywords

classical ergodic theorylow dimensional dynamicsnumber theory

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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