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

Ian D. Morris
2017 Annales Academiae Scientiarum Fennicae: Mathematica  
We investigate a formula of Falconer which describes the typical value of the generalised Rényi dimension, or generalised q-dimension, of a self-affine measure in terms of the linear components of the affinities. We show that in contrast to a related formula for the Hausdorff dimension of a typical self-affine set, the value of the generalised q-dimension predicted by Falconer's formula varies discontinuously as the linear parts of the affinities are changed. Conditionally on a conjecture of
more » ... a conjecture of Bochi and Fayad, we show that the value predicted by this formula for pairs of two-dimensional affine transformations is discontinuous on a set of positive Lebesgue measure. These discontinuities derive from discontinuities of the lower spectral radius which were previously observed by the author and Bochi. s d , s ≥ d, https://doi.org/10.5186/aasfm.2017.4214 2010 Mathematics Subject Classification: Primary 28A80.
doi:10.5186/aasfm.2017.4214 fatcat:e5i4i6ifx5hfrg4gnsfg62dffe