On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$

TL;DR

This paper investigates the structure and dimensions of self-similar sets with overlaps in , providing an inverse entropy theorem and classifying possible behaviors under certain linear and affine conditions.

Contribution

It introduces an inverse theorem for entropy growth of convolutions in  and characterizes self-similar sets with overlaps, extending to smooth Lie group actions.

Findings

Dimension equals the minimum of  and the similarity dimension in many cases

Existence of super-exponentially close compositions of the defining maps

Identification of invariant subspaces under the linearization of the iterated function system

Abstract

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, we show 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. We give 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.