Dimension of generic self-affine sets with holes

Let ( Σ , σ ) be a dynamical system, and let U ⊂ Σ . Consider the survivor set Σ U = x ∈ Σ ∣ σ n ( x ) ∉ U for all n of points that never enter the subset U . We study the size of this set in the case when Σ is the symbolic space associated to a self-affine set Λ , calculating the dimension of the p...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2019-03, Vol.188 (3), p.527-546
Main Authors: Koivusalo, Henna, Rams, Michał
Format: Article
Language:English
Subjects:
Asymptotic methods
Cylinders
Mathematics
Mathematics and Statistics
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Summary:Let ( Σ , σ ) be a dynamical system, and let U ⊂ Σ . Consider the survivor set Σ U = x ∈ Σ ∣ σ n ( x ) ∉ U for all n of points that never enter the subset U . We study the size of this set in the case when Σ is the symbolic space associated to a self-affine set Λ , calculating the dimension of the projection of Σ U as a subset of Λ and finding an asymptotic formula for the dimension in terms of the Käenmäki measure of the hole as the hole shrinks to a point. Our results hold when the set U is a cylinder set in two cases: when the matrices defining Λ are diagonal; and when they are such that the pressure is differentiable at its zero point, and the Käenmäki measure is a strong-Gibbs measure.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-018-1187-6