Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 63-88

DIMENSION OF SELF-AFFINE SETS WITH HOLES

Andrew Ferguson, Thomas Jordan and Michal Rams

D01 Du Cane Ct, Balham High Rd, London, SW17 7JQ, U.K.; arjferguson 'at' gmail.com

University of Bristol, School of Mathematics
University Walk, Bristol, BS8 1TW, U.K.; thomas.jordan 'at' bris.ac.uk

Polish Academy of Sciences, Institute of Mathematics
ul. Sniadeckich 8, 00-956 Warszawa, Poland; m.rams 'at' impan.gov.pl

Abstract. In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let X ⊆ T² denote a Bedford–McMullen set and T : X → X the natural expanding toral endomorphism which leaves X invariant. For an open set U ⊂ X we let

X^U = {x ∈ X : T^k(x) ∉ U for all k}.

We investigate the box and Hausdorff dimensions of X_U for both a fixed Markov hole and also when U is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through U, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.

2010 Mathematics Subject Classification: Primary 37C45, 37C30.

Key words: Self-affine sets, escape rate, transfer operator.

Reference to this article: A. Ferguson, T. Jordan and M. Rams: Dimension of self-affine sets with holes. Ann. Acad. Sci. Fenn. Math. 40 (2015), 63-88.

Full document as PDF file

doi:10.5186/aasfm.2015.4007