Mixmaster Universe in a 2D non-commutative GUP framework

In this work, we examine the dynamical aspects of the cosmological Mixmaster model within the framework of a non-commutative Generalized Uncertainty principle (GUP) theory. The theory is formulated classically by introducing a well-defined symplectic form that differs from the ordinary one, thereby inducing a deformation of the Poisson brackets. We first investigate the behavior of the Bianchi I and Bianchi II models using Misner variables. Then, we study the Bianchi IX model in the Mixmaster approximation, which is well-known for accurately reproducing the dynamics of the point-particle Universe approaching the cosmological singularity. We derive the corresponding Belinsky-Khalatnikov-Lifshitz (BKL) map and explore its resulting features, shaped by the effects of the non-commutative GUP scheme.