Discontinuous maps exhibiting symmetry Lebesgue-almost everywhere

Abstract

In this paper we analyse the existence of a certain type of symmetry in the context of discontinuous maps. The classical notions of symmetry cannot be applied due to the existence of discontinuities and a broader version using a measure-theory perspective is introduced. We show that a group structure is also present under the new type of symmetry and derive results which are analogous in nature to results in the theory of continuous maps. Our motivation stems from examples of symmetric patterns arising in simulations with the Goetz map.