A non-Borel special alpha-limit set in the square

TL;DR

This paper investigates the complexity of special alpha-limit sets in non-invertible dynamical systems, demonstrating they are always analytic but not necessarily Borel, even for surjective maps on the unit square.

Contribution

It proves that special alpha-limit sets are always analytic but can be non-Borel, resolving an open question in the field.

Findings

Special alpha-limit sets are always analytic.

Such sets are not necessarily Borel.

This holds even for surjective maps on the unit square.

Abstract

We consider the complexity of special $\alpha$-limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.