Self-affine sets in analytic curves and algebraic surfaces

Ding, Feng; Käenmäki, Antti

doi:10.5186/aasfm.2018.4306

Cited by 7 publications

(5 citation statements)

References 5 publications

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“…As before, if an affine iterated function system preserves an affine subspace with dimension smaller than the affinity dimension then the affine subspace necessarily contains the attractor, so the dimension of the subspace bounds the dimension of the attractor and the iterated function system is necessarily exceptional. In the affine case a further possibility exists that the iterated function system may lack an invariant affine subspace but preserve a low-dimensional variety or manifold, a phenomenon which is investigated in [2,10]. For example, if T 1 , .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On dense intermingling of exact overlaps and the open set condition

Morris¹

2022

Preprint

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We prove that certain families of homogenous affine iterated function systems in R d have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of R d with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. Our result encompasses a one-dimensional example due to Kenyon which arises as a special case.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On dense intermingling of exact overlaps and the open set condition

Morris¹

2022

Preprint

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“…As before, if an affine iterated function system preserves an affine subspace with dimension smaller than the affinity dimension then the affine subspace necessarily contains the attractor, so the dimension of the subspace bounds the dimension of the attractor and the iterated function system is necessarily exceptional. In the affine case, a further possibility exists that the iterated function system may lack an invariant affine subspace but preserve a low-dimensional variety or manifold, a phenomenon which is investigated in [2, 10]. For example, if are contractions of the form then the two-dimensional variety is preserved by the iterated function system , hence this variety contains the attractor and the dimension of the attractor cannot be higher than two.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On dense intermingling of exact overlaps and the open set condition

Morris

2022

Proceedings of the Edinburgh Mathematical Society

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We prove that certain families of homogenous affine iterated function systems in $\mathbb {R}^{d}$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan–Pollicott–Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb {R}^{d}$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. This result encompasses a one-dimensional example due to Kenyon which arises as a special case.

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“…Therefore, y = x ∈ (V + y) ∩ X which is a contradiction. We remark that in general it is not possible to choose ε = 0 above as a planar self-affine set can be contained for example in a parabola; see [5,26]. The following proposition gives the upper bound in Theorem 3.1.…”

Section: Lower Dimension Of Self-affine Setsmentioning

confidence: 99%

Assouad dimension of planar self-affine sets

Bárány

Käenmäki

Rossi

2020

Trans. Amer. Math. Soc.

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We calculate the Assouad dimension of a planar self-affine set X satisfying the strong separation condition and the projection condition and show that X is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set X adheres to very strong tangential regularity by showing that any two points of X, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.

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Self-affine sets in analytic curves and algebraic surfaces

Abstract: We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces do not contain non-trivial self-affine sets.

Cited by 7 publications

References 5 publications

On dense intermingling of exact overlaps and the open set condition

On dense intermingling of exact overlaps and the open set condition

On dense intermingling of exact overlaps and the open set condition

Assouad dimension of planar self-affine sets

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