On Falconer’s Formula for the Generalized Rényi Dimension of a Self-affine Measure
نویسنده
چکیده
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.
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تاریخ انتشار 2017