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

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.