Self-affine sets in analytic curves and algebraic surfaces

TL;DR

This paper characterizes which analytic curves contain non-trivial self-affine sets and proves that compact algebraic surfaces do not contain such sets, advancing understanding of self-affinity in geometric structures.

Contribution

It provides a complete characterization of analytic curves with self-affine sets and establishes that compact algebraic surfaces cannot host non-trivial self-affine sets.

Findings

Analytic curves containing self-affine sets are fully characterized.

Compact algebraic surfaces are proven not to contain non-trivial self-affine sets.

Abstract

We characterize analytic curves that contain non-trivial self-affine sets. We also prove that compact algebraic surfaces cannot contain non-trivial self-affine sets.