Scaling scenery of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mo stretchy="false">(</mml:mo><mml:mo>×</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mo>×</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> invariant measures

Ferguson, Andrew; Fraser, Jonathan M.; Sahlsten, Tuomas

doi:10.1016/j.aim.2014.09.019

Cited by 27 publications

(42 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Of course, it is enough to assume that A contains a set of Hausdorff and packing dimension s > 1. With this observation, this theorem recovers, generalizes and improves the results on distance sets from [15,6,16,20], again outside of the endpoint. We also point out that packing dimension is smaller than upper box counting (Minkowski) dimension, so equality of Hausdorff and box dimension also implies the conclusion of the theorem.…”

supporting

confidence: 71%

On the Hausdorff dimension of pinned distance sets

Shmerkin

2019

Isr. J. Math.

View full text Add to dashboard Cite

We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension 1 (in particular, for many x ∈ A). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint s = 1.2010 Mathematics Subject Classification. Primary: 28A75, 28A80; Secondary: 49Q15.

show abstract

supporting

confidence: 71%

On the Hausdorff dimension of pinned distance sets

Shmerkin

2019

Isr. J. Math.

View full text Add to dashboard Cite

show abstract

“…To make the connection, we introduce the Euclidean version of CP distributions, which are closely related to symbolic CP distributions. In this section, we partially follow [14, Section 2.1]. We introduce the theory only in

R^{2}

, where we shall use it.…”

Section: Preliminariesmentioning

confidence: 99%

“…Consider the compact metric space

T \times {[n_{1}]}^{double-struckN} \times {[n_{2}]}^{double-struckN}

. Define a map

Z : T \times {[n_{1}]}^{double-struckN} \times {[n_{2}]}^{double-struckN} \to T \times {[n_{1}]}^{double-struckN} \times {[n_{2}]}^{double-struckN}

by taking

Z false(t, ω, η false) = (R_{θ} false(t false), σ_{t} false(ω false), σ false(η false)) .

The following proposition was proved during the proof of [14, Proposition 4.3]. Theorem Let

α_{1}, α_{2}

be Bernoulli measures on

{false[n_{1} false]}^{N}

{false[n_{2} false]}^{N}

, respectively.…”

Section: A Cp Chain On the Space Of Slices Of A Family Of Product Setsmentioning

confidence: 99%

See 1 more Smart Citation

Slicing theorems and rigidity phenomena for self‐affine carpets

Algom

2020

Proc. London Math. Soc.

View full text Add to dashboard Cite

Let F be a Bedford–McMullen carpet defined by independent exponents. We prove that dim¯Bfalse(ℓ∩Ffalse)⩽maxfalse{dim∗F−1,0false} for all lines ℓ not parallel to the principal axes, where dim∗ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford–McMullen carpets, that is, carpets F and E such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self‐affine measures on them are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing theorem for products of certain Cantor sets. This theorem is a generalization of the results of Shmerkin [Ann. of Math. (2) 189 (2019) 319–391] and Wu [Ann. of Math. (2) 189 (2019) 707–751], which proved Furstenberg's slicing conjecture [Problems in analysis (ed. R. C. Gunning; Princeton University Press, Princeton, NJ, 1970) 41–59].

show abstract

“…carpets (see [13,14,25]). Another method of study and a class of self-affine sets to which it applies was introduced by Falconer [8] and later extended by Solomyak [32].…”

Section: Introductionmentioning

confidence: 99%

Dimension of self-affine sets for fixed translation vectors

Bárány

Käenmäki

Koivusalo

2018

J. London Math. Soc.

View full text Add to dashboard Cite

An affine iterated function system (IFS) is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self‐affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self‐affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self‐affine measures. In this article, we have an orthogonal approach. We introduce a class of self‐affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier–Young theory that was recently verified for affine IFSs by Bárány and Käenmäki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self‐affine sets and measures in any Euclidean space.

show abstract

Scaling scenery of (×m,×n) invariant measures

Cited by 27 publications

References 31 publications

On the Hausdorff dimension of pinned distance sets

On the Hausdorff dimension of pinned distance sets

Slicing theorems and rigidity phenomena for self‐affine carpets

Dimension of self-affine sets for fixed translation vectors

Contact Info

Product

Resources

About