In this note we give generalisations of Livsic’s result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) β-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately...

Let $T$ denote a positive operator with spectral radius $1$ on, say, an $L^p$-space. A classical result in infinite dimensional Perron--Frobenius theory says that, if is irreducible and power bounded, then its peripheral point spectrum either empty or subgroup of the unit circle. In this note we show that analogous assertion for entire fails. More precisely, every finite union $U$ subgroups c...