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

• Verfasst von: Hochman, Mike [VerfasserIn]
• Titel: On self-similar sets with overlaps and inverse theorems for entropy in Rd
• Verf.angabe: Mike Hochman
• Verlagsort: Providence
• Verlag: American Mathematical Society
• E-Jahr: 2020
• Jahr: [2020]
• Umfang: 1 Online-Ressource (v, 100 Seiten)
• Gesamttitel/Reihe: Memoirs of the American Mathematical Society ; volume 265, number 1287
• Fussnoten: Im Titel ist das kleine d hochgestellt ; Description based on publisher supplied metadata and other sources
• ISBN: 978-1-4704-6142-3
• Datenträger: Online-Ressource
• Sprache: eng
• Bibliogr. Hinweis: Erscheint auch als : Druck-Ausgabe: Hochman, Mike: On self-similar sets with overlaps and inverse theorems for entropy in R^d. - Providence, RI : American Mathematical Society, 2020. - pages cm
• Sach-SW: Electronic books
• K10plus-PPN: 172573415X
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

The author studies self-similar sets and measures on \mathbb{R}^{d}. Assuming that the defining iterated function system \Phi does not preserve a proper affine subspace, he shows that one of the following holds: (1) the dimension is equal to the trivial bound (the minimum of d and the similarity dimension s); (2) for all large n there are n-fold compositions of maps from \Phi which are super-exponentially close in n; (3) there is a non-trivial linear subspace of \mathbb{R}^{d} that is preserved by the linearization of \Phi and whose translates typically meet the set or measure in full dimension. In particular, when the linearization of \Phi acts irreducibly on \mathbb{R}^{d}, either the dimension is equal to \min\{s,d\} or there are super-exponentially close n-fold compositions. The author gives a number of applications to algebraic systems, parametrized systems, and to some classical examples. The main ingredient in the proof is an inverse theorem for the entropy growth of convolutions of measures on \mathbb{R}^{d}, and the growth of entropy for the convolution of a measure on the orthogonal group with a measure on \mathbb{R}^{d}. More generally, this part of the paper applies to smooth actions of Lie groups on manifolds.

Abstract

Cover -- Title page -- Chapter 1. Introduction -- 1.1. Setup: Self-similar sets and measures -- 1.2. Main results -- 1.3. Parametric families -- 1.4. Applications -- 1.5. Organization and notation -- Acknowledgment -- Chapter 2. An inverse theorem for the entropy of convolutions -- 2.1. Entropy and additive combinatorics -- 2.2. Concentration and saturation on subspaces -- 2.3. Component measures -- 2.4. An inverse theorem for convolutions on ℝ^{ } -- 2.5. An inverse theorem for isometries acting on ℝ^{ } -- 2.6. Generalizations -- Chapter 3. Entropy, concentration, uniformity and saturation -- 3.1. Preliminaries on entropy -- 3.2. Global entropy from local entropy -- 3.3. First lemmas on concentration, uniformity, saturation -- 3.4. Concentration and saturation of components -- 3.5. The space of subspaces -- 3.6. Geometry of thickened subspaces -- 3.7. Minimally concentrated and maximally saturated subspaces -- 3.8. Measures with uniformly concentrated components -- Chapter 4. The inverse theorem in ℝ^{ } -- 4.1. Elementary properties of convolutions -- 4.2. Mean, covariance and concentration -- 4.3. Gaussian measures and the Berry-Esseen-Rotar estimate -- 4.4. Multi-scale analysis of repeated self-convolutions -- 4.5. The Kaĭmanovich-Vershik lemma -- 4.6. Proof of the inverse theorem -- Chapter 5. Inverse theorem for the action of the isometry group on ℝ^{ } -- 5.1. Concentration and saturation on random subspaces -- 5.2. From concentration of Euclidean components to ₀-components -- 5.3. Entropy and the ₀-action on ℝ^{ } -- 5.4. Linearization of the ₀-action -- 5.5. Proof of the inverse theorem -- 5.6. Generalizations -- Chapter 6. Self-similar sets and measures on ℝ^{ } -- 6.1. Almost-invariance and invariance -- 6.2. Saturation at level -- 6.3. Saturated subspaces of self-similar measures.