Classifying the invariant measures for a given dynamical system is a fundamental problem.

In the field of homogeneous dynamics, several important theorems give us an essentially complete picture.Moving away from homogeneous dynamics — results are scarcer, mainly due to some profound difficulties carrying out the techniques used in homogeneous dynamics.

A recent development in Teichmuller dynamics — the celebrated magic wand theorem of Eskin–Mirzakhani, gives one such example and actually provides a technique — the factorization method — for proving such results in certain systems.

I will explain how one can implement the factorization method of Eskin–Mirzakhani in smooth dynamics, in order to achieve measure classification of u-Gibbs states for non-integrable Anosov actions. Moreover, I will try to explain some applications of the theorem, including a result of Avila–Crovosier–Eskin–Potrie–Wilkinson–Zhang towards Gogolev’s conjecture on actions on the 3D torus.