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

Morris, ID (2017) On Falconer's formula for the generalized Rényi dimension of a self-affine measure Annales Academiae Scientiarum Fennicae.

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Abstract

We investigate a formula of K. Falconer which describes the typical value of the generalised R´enyi dimension, or generalised q-dimension, of a selfaffine 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 J. Bochi and B. 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 J. Bochi.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Morris, IDUNSPECIFIEDUNSPECIFIED
Date : 9 February 2017
Identification Number : 10.5186/aasfm.00
Copyright Disclaimer : Copyright © Academia Scientiarum Fennica
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 09 Aug 2016 09:21
Last Modified : 09 Aug 2016 09:21
URI: http://epubs.surrey.ac.uk/id/eprint/811633

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