The beta-map for β > 1 is a simple piecewise linear expanding map on the unit interval and its ergodic properties can be investigated via its Perron-Frobenius operator, which is a bounded linear operator defined on the space of functions of bounded variation. On that space, the Perron-Frobenius operator is to be quasi-compact, i.e., any spectrum whose modulus is greater than 1/β is an isolated eigenvalue with finite multiplicity. In particular, it has 1 as its leading eigenvalue, although the less are known for the other isolated eigenvalues (non-leading eigenvalues), including their existence.
In this talk, we see that the set of β’s such that the Perron-Frobenius operator corresponding to β has at least one non-leading eigenvalue is open and dense in (1, +∞) and that each non-leading eigenvalue is continuous but non-differentiable as a function of β defined on that set. Furthermore, we establish the Hölder exponent of that function for each non-leading eigenvalue.