Self-affine 2-attractors and tiles

TL;DR

This paper classifies self-affine 2-attractors in Euclidean spaces, analyzing their topological and spectral properties, and estimates their diversity using algebraic and number-theoretic tools, with extensions to multi-digit cases.

Contribution

It provides a complete classification of isotropic 2-attractors and links their structure to the spectrum of dilation matrices, extending understanding of self-affine tiles.

Findings

All isotropic 2-attractors are homeomorphic but not diffeomorphic.

A 2-attractor is uniquely determined by the spectrum of its dilation matrix.

The number of 2-attractors is estimated via Mahler measure of associated polynomials.

Abstract

We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In the general, non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to an affine similarity, by the spectrum of the dilation matrix. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with the free coefficient $\pm 2$. The total number of such polynomials is estimated by the Mahler measure. We present several infinite series of such polynomials. For some of the 2-attractors, their H\"older exponents are found. Some of our results are extended to attractors with an arbitrary number of digits.