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We investigate a formula of K. Falconer which describes the typical value ofthe generalised R\'enyi dimension, or generalised $q$-dimension, of aself-affine measure in terms of the linear components of the affinities. Weshow that in contrast to a related formula for the Hausdorff dimension of atypical self-affine set, the value of the generalised $q$-dimension predictedby Falconer's formula varies discontinuously as the linear parts of theaffinities are changed. Conditionally on a conjecture of J. Bochi and B. Fayad,we show that the value predicted by this formula for pairs of two-dimensionalaffine transformations is discontinuous on a set of positive Lebesgue measure.These discontinuities derive from discontinuities of the lower spectral radiuswhich were previously observed by the author and J. Bochi.

On Falconer's formula for the generalised Renyi dimension of a self-affine measure