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On self-similar sets with overlaps and inverse theorems for entropy.(English)Zbl 1337.28015

Let \(\Phi :=\{\varphi_i : i\in \Lambda\}\) be a finite collection of linear contractions of the form \(\varphi_i (x) := r_i x + a_i\), where \(|r_i| < 1\) and \(a_i\in \mathbb R\). Let \(X\neq\emptyset\) be the attractor of the iterated function system \(\Phi\) and let the self-similar measure associated with \(\Phi\) and a probability vector \((p_i)_{i\in \Lambda}\) be denoted by \(\mu\). The author investigates the Hausdorff dimension \(\dim \mu\) of \(\mu\) in case the images \(\varphi_i X\) have significant overlap. The main result is the following: If \(\dim\mu < \min\{1, s\}\), where \(s\) is the similarity dimension of \(X\), then there exist superexponentially close cylinders at small enough scales. As a consequence of this result, the Furstenberg conjecture on the projections of the one-dimensional Sierpiński gasket is proven and some progress on the Bernoulli convolutions problem is achieved.

MSC:

28A80 Fractals
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
37C35 Orbit growth in dynamical systems
28D20 Entropy and other invariants

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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