The matrix potential game and structures of self-affine sets

TL;DR

This paper introduces a new potential game framework to analyze self-affine sets, establishing their non-emptiness, dimension bounds, intersection properties, and the presence of finite configurations, advancing the understanding of fractal structures.

Contribution

The paper develops a novel potential game approach to study self-affine sets, providing new results on their structure, dimension, and finite configurations, complementing existing self-similar set research.

Findings

Self-affine sets are winning in the new potential game.

Sets with strong winning conditions have positive Hausdorff dimension.

Self-affine sets contain homothetic copies of finite sets under certain conditions.

Abstract

We present a new variant of the potential game and show that certain compact subsets of $\R^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \N$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets.