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 -