A self-similar measure with dense rotations, singular projections and discrete slices

Rapaport, Ariel

doi:10.1016/j.aim.2017.10.007

Cited by 7 publications

(3 citation statements)

References 13 publications

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“…The extension of Theorem 1 in various directions has been an active topic of research since its original publication. One major area of research has been the problem of understanding systematically what happens when the open set condition is removed (such as in [2,14,28,29,37,43,46,48]) and this line of research has focused especially on the dimensions of the resulting measures as opposed to the resulting sets. A second major direction of extension of Theorem 1 is that in which the transformations T i are allowed to be arbitrary affine contractions instead of similitudes: this line of research dates back to the work of Bedford, McMullen and Falconer in the 1980s [7,20,38] and has been particularly active within the last few years (see for example [4,5,6,11,17,22,25,35,44]).…”

Section: Introductionmentioning

confidence: 99%

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

Morris¹,

Sert²

2019

Preprint

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A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier work in the thermodynamic formalism of affine iterated function systems due to Feng, Käenmäki, Bochi and the first named author and also relies on the work of Benoist and Quint on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

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Section: Introductionmentioning

confidence: 99%

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

Morris¹,

Sert²

2019

Preprint

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“…In case the self-similar measure is generated by maps having no rotations, the proposition is new in the non-homogeneous case, and also relaxes the separation assumption compared to [9,Theorem B(i)] in the homogeneous case. If the maps have dense rotations, then the reader is referred to the works of Shmerkin and Solomyak [9, Theorem B(ii)] and Rapaport [6].…”

Section: Introductionmentioning

confidence: 99%

Absolute continuity in families of parametrised non-homogeneous self-similar measures

Käenmäki¹,

Orponen²

2018

Preprint

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In 2016, Shmerkin and Solomyak showed that if U Ă R is an interval, and tµuuuPU is an analytic family of homogeneous self-similar measures on R with similitude dimensions exceeding one, then, under a mild transversality assumption, µu ! L 1 for all parameters u P U zE, where dimH E " 0. The purpose of this paper is to generalise the result of Shmerkin and Solomyak to non-homogeneous self-similar measures. As a corollary, we obtain new information about the absolute continuity of projections of nonhomogeneous planar self-similar measures. CONTENTS 24 4.3. Concluding the proof of the main theorem 27 Appendix A. Order K transversality and the size of exceptions 28 References 29

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“…Taking these results into account, one might expect the sets E, defined above, and F = {z ∈ S : ν is not DC with respect to P z } to be empty whenever the dimension of ν exceeds 1. Nevertheless, in [Rap2] the author has constructed an example, of a measure ν as above, for which E and F are nonempty and even residual. Moreover, in this example there exists a G δ -subset of directions z ∈ S for which ν z,w is discrete, and hence has dimension 0, for ν-a.e.…”

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confidence: 99%

On self-similar measures with absolutely continuous projections and dimension conservation in each direction

Rapaport¹

2019

Ergod. Th. Dynam. Sys.

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Relying on results due to Shmerkin and Solomyak, we show that outside a 0-dimensional set of parameters, for every planar homogeneous selfsimilar measure ν, with strong separation, dense rotations and dimension greater than 1, there exists q > 1 such that {Pzν} z∈S ⊂ L q (R). Here S is the unit circle and Pzw = z, w for w ∈ R 2 . We then study such measures. For instance, we show that ν is dimension conserving in each direction and that the map z → Pzν is continuous with respect to the weak topology of L q (R).

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A self-similar measure with dense rotations, singular projections and discrete slices

Cited by 7 publications

References 13 publications

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

Absolute continuity in families of parametrised non-homogeneous self-similar measures

On self-similar measures with absolutely continuous projections and dimension conservation in each direction

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