We prove that the Frobenius--Perron operator U of the cusp map
F:[−1,1]→[−1,1], F(x)=1−2∣x∣ (which is an approximation of the
Poincar\'e section of the Lorenz attractor) has no analytic eigenfunctions
corresponding to eigenvalues different from 0 and 1. We also prove that for any
q∈(0,1) the spectrum of U in the Hardy space in the disk
\{z\in\C:|z-q|<1+q\} is the union of the segment [0,1] and some finite or
countably infinite set of isolated eigenvalues of finite multiplicity.Comment: Submitted to JMP; The description of the spectrum in some Hardy
spaces is adde