TY - JOUR

T1 - Eigenfunctions of the Perron–Frobenius operators for generalized beta-maps

AU - Suzuki, Shintaro

N1 - Funding Information:
This work was supported by JSPS [KAKENHI/ 20K14331]. The author would like to express sincere thanks to Professor T. Morita and Professor H. Takahasi for their valuable comments. The author would also like to thank the anonymous referee for helpful suggestions.
Publisher Copyright:
© 2021 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2022

Y1 - 2022

N2 - For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1.

AB - For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1.

KW - dynamical zeta functions

KW - eigenspaces

KW - Lasota–Yorke maps

KW - Perron–Frobenius operators

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U2 - 10.1080/14689367.2021.1998378

DO - 10.1080/14689367.2021.1998378

M3 - Article

AN - SCOPUS:85119995617

SN - 1468-9367

VL - 37

SP - 9

EP - 28

JO - Dynamical Systems

JF - Dynamical Systems

IS - 1

ER -