  The Hilbert space $\mathcal H$ of backward renormalisation of an anyonic
quantum spin chain affords a unitary representation of Thompson's group $F$ via
local scale transformations. Given a vector in the canonical dense subspace of
$\mathcal H$ we show how to calculate the corresponding spectral measure for
any element of $F$ and illustrate with some examples. Introducing the
"essential part" of an element we show that the spectral measure of any vector
in $\mathcal H$ is, apart from possibly finitely many eigenvalues, absolutely
continuous with respect to Lebesgue measure. The same considerations and
results hold for the Brown-Thompson groups $F_n$ (for which $F=F_2$).
