Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 227-238

ON FALCONER'S FORMULA FOR THE GENERALIZED RÉNYI DIMENSION OF A SELF-AFFINE MEASURE

Ian D. Morris

University of Surrey, Department of Mathematics
Guildford GU2 7XH, U.K.; i.morris 'at' surrey.ac.uk

Abstract. 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 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.

2010 Mathematics Subject Classification: Primary 28A80.

Key words: Lower spectral radius, self-affine measure, self-affine set, q-dimension, Rényi dimension.

Reference to this article: I. D. Morris: On Falconer's formula for the generalised Rényi dimension of a self-affine measure. Ann. Acad. Sci. Fenn. Math. 42 (2017), 227-238.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4214