NZMRI 2026

Title: Measure Rigidity in Smooth Dynamics and the study of u-Gibbs States

Abstract: 
Roughly, measure rigidity asks the following: given a measure, when can we expect it to have ‘extra' structure? Answers to this type of question are of central interest in dynamics, and a core component to the work of Ratner, Benoist-Quint, Eskin-Mirzakhani, Brown-Eskin-Filip-Rodriguez Hertz, and many others. In the first lecture, we will lay the groundwork for why measure rigidity is interesting, and discuss several consequences of measure classification, i.e., orbit classification theorems, applications to number theory, geometry, etc. 

In the second lecture, we will focus in on the case of physical measures. Physical measures are an important tool in the study of hyperbolic dynamics, governing, for example, the statistical properties of the orbit of almost every point with respect to volume (in the dissipative setting). The well-studied uniformly hyperbolic (Anosov) diffeomorphisms and flows always have ergodic physical measures, whereas the more general class of partially hyperbolic systems lose this property. For these systems, we are instead guaranteed the existence of at least one, and possibly infinitely many, ergodic u-Gibbs measure(s). In the case of a unique u-Gibbs measure, that measure is automatically physical. 

Thus, a natural question in the partially hyperbolic setting is the following: under what conditions is there a unique u-Gibbs measure? More generally, which u-Gibbs measures are physical? This question was partially answered in dimension three by Eskin, Potrie, and Zhang. We will discuss an extension of the result of Eskin-Potrie-Zhang to arbitrary dimensions, and focus on the dichotomy that arises: roughly, a u-Gibbs measure is physical if and only if it is not jointly integrable of some order.