On self-similar sets with overlaps and inverse theorems for entropy

By Michael Hochman. 2012

338 citations recorded by the source corpus.

We study the dimension of self-similar sets and measures on the line. We show that if the dimension is less than the generic bound of minf1;sg, where s is the similarity dimension, then there are superexponentially close cylinders at all small enough scales. This is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients. As applications we prove Furstenberg’s conjecture on projections of the one-dimensional Sierpinski gasket and achieve some progress on the Bernoulli convolutions problem and, more generally, on problems about parametric families of self-similar measures. The key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef64a70101a8b6d93a87f4b72fdbe5ee940de14b

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Citations in the corpus

Sixty years of fractal projections (2014, background): “Hochman and Shmerkin [35] proved the theorem in higher dimensions, including the extension to C1 mappings, for E satisfying the open set condition.”
New Bounds on the Dimensions of Planar Distance Sets (2018, methodology): “We emphasize that this method to bound the dimension of (linear or nonlinear) projections from below goes back in various forms to [10, 9, 22], although the use of projected L norms (rather than projected entropies) was first used in [26].”
On the Assouad dimension of self-similar sets with overlaps (2014, background): “Hochman’s arguments rely on a careful application of ergodic theory and in establishing inverse theorems for the growth of certain entropy functions.”
On the modulus of continuity for spectral measures in substitution dynamics (2013, methodology): “Having dimension one for a measure falls just short of absolute continuity, but even more recently Shmerkin [37] combined Hochman’s Theorem with the Erdős-Kahane result mentioned above to obtain absolute continuity outside a zero-dimensional set of exceptions.”
Absolute continuity of Bernoulli convolutions for algebraic parameters (2016, methodology): “In comparison, Hochman’s argument prove (under milder conditions) H(µλ; 2 −n|2−n+1) → 1, which is enough to conclude that dimµλ = 1.”
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions (2013, background): “While Hochman’s method does not directly give any result on absolute continuity (see [10, Section 1.5]), in this paper we use his result to obtain a short proof of the following: Theorem 1.1.”
On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems (2016, background): “Similar to the results of [21, 22] that we rely on, we can also show that in fairly general parametrized families, there is equality of Hausdorff and affinity dimension outside of a co-dimension 1 set of parameters.”
Equidistribution from fractal measures (2013, background): “In the non-uniformly contracting case a minor (but not short) modification of the arguments in [26] is needed; this will appear separately.”
Recent progress on Bernoulli convolutions (2016, methodology): “We present a mixture of ideas from the papers [29], [61] and [16].”
On the dimension of Bernoulli convolutions for all transcendental parameters (2018, background): “Hochman proved in another landmark paper [12] that the set E is of 0 packing and Hausdorff dimension.”
On restricted families of projections in ℝ3 (2013, methodology): “Hochman [3], but the methods of proof are very different.”
Absolute continuity of self-similar measures, their projections and convolutions (2014, methodology): “We first state a result that follows from Hochman’s work [11, 12]; notice that it is very close to the statement of Theorem A, except that the conclusion is about Hausdorff dimension rather than absolute continuity.”
Hausdorff dimension of planar self-affine sets and measures (2017, neutral): “(ii) Applying the multidimensional inverse theorem The inverse theorem in R from [16] says that in order for a convolution τ ∗ ν of measures in R(2) to have entropy that is essentially the same as that of ν alone, it must be the case that, at most scales δ, there is a linear subspace V = Vδ ≤ R(2) s”