The Hausdorff dimension of the projections of self-affine carpets

Ferguson, Andrew; Jordan, Thomas; Shmerkin, Pablo

doi:10.4064/fm209-3-1

Cited by 21 publications

(24 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It suffices to prove that for any ε > 0 we can find a subset of F which has lower dimension within ε of the Hausdorff dimension. Via an elegant application of Stirling's formula, Ferguson-Jordan-Shmerkin [FJS,Lemma 4.3] showed that one can always find a subset of such a carpet generated by a subsystem of an iterate of the original IFS which both has Hausdorff dimension within ε of the original Hausdorff dimension and has the key additional property that the fibres are uniform. Then it follows from [Fr,Theorem 2.13] that the lower dimension and the Hausdorff dimension of the subset coincide, thus proving the result.…”

Section: Self-affine Carpetsmentioning

confidence: 99%

Assouad-type spectra for some fractal families

Fraser¹,

Yu²

2018

Indiana Univ. Math. J.

View full text Add to dashboard Cite

In a previous paper we introduced a new 'dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers.2010 Mathematics Subject Classification. Primary: 28A80. Secondary: 37C45, 82B43.

show abstract

Section: Self-affine Carpetsmentioning

confidence: 99%

Assouad-type spectra for some fractal families

Fraser¹,

Yu²

2018

Indiana Univ. Math. J.

View full text Add to dashboard Cite

show abstract

“…Proof. Let Γ be the set constructed by Proposition 4.3 with ε/2 in place of ε, and with n chosen large enough that e −nε/2 < 1 16 . We will find an integer n > n and a set Γ ⊆ {1, .…”

Section: Regular Subsystemsmentioning

confidence: 99%

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

Morris¹,

Shmerkin²

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Bárány, HochmanSolomyak and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any domination assumptions.

show abstract

“…Early results of this type for sets were obtained in [3,21,11]. Recall that the (lower) Hausdorff dimension of a measure µ is…”

Section: Introductionmentioning

confidence: 99%

L^qdimensions and projections of random measures

et al. 2016

Self Cite

View full text Add to dashboard Cite

We prove preservation of L q dimensions (for 1 < q ≤ 2) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures supported on 1-variable fractals as special cases. We prove a similar result for certain convolutions, extending a result of Nazarov, Peres and Shmerkin. Recently many related results have been obtained for Hausdorff dimension, but much less is known for L q dimensions.2000 Mathematics Subject Classification. Primary 28A80, Secondary 28A78, 37H99.[8]). In general, q → D q may be strictly decreasing (this is a reflection of the multifractality of µ), but it may also be constant. For example, if µ is Ahlforsregular with exponent d (that is, if C −1 r d ≤ µ(B(x, r)) ≤ C r d for all x ∈ supp(µ)) then D q µ = dim H µ = d for all q. For many measures of dynamical origin, such as self-similar measures, the limit in the definition of D q is known to exist, see [22].The only previous result on L q dimensions of projected measures was obtained in [19]. There it is proved that if µ, ν are self-similar measures satisfying certain natural assumptions, then for any q ∈ (1, 2], D q (Π(µ × ν)) = min(D q (µ × ν), 1)

show abstract

The Hausdorff dimension of the projections of self-affine carpets

Cited by 21 publications

References 11 publications

Assouad-type spectra for some fractal families

Assouad-type spectra for some fractal families

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

L^qdimensions and projections of random measures

Contact Info

Product

Resources

About

The Hausdorff dimension of the projections of self-affine carpets

Cited by 21 publications

References 11 publications

Assouad-type spectra for some fractal families

Assouad-type spectra for some fractal families

On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

Lqdimensions and projections of random measures

Contact Info

Product

Resources

About

L^qdimensions and projections of random measures