Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 313-324

THE DIMENSIONS OF INHOMOGENEOUS SELF-AFFINE SETS

Stuart A. Burrell and Jonathan M. Fraser

University of St Andrews, School of Mathematics and Statistics
St Andrews, KY16 9SS, UK; sb235 'at' st-andrews.ac.uk

University of St Andrews, School of Mathematics and Statistics
St Andrews, KY16 9SS, UK; jmf32 'at' st-andrews.ac.uk

Abstract. We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets.

2010 Mathematics Subject Classification: Primary 28A80.

Key words: Inhomogeneous attractor, self-affine set, box dimension, affinity dimension.

Reference to this article: S. A. Burrell and J. M. Fraser: The dimensions of inhomogeneous self-affine sets. Ann. Acad. Sci. Fenn. Math. 45 (2020), 313-324.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4516