Cartan Actions of Higher Rank Abelian Groups and their Classification

Abstract

We study R k actions with a dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. When k ≥ 2, we classify totally Cartan actions up to smooth conjugacy under mild transitivity assumptions. These are the first classification results for k = 2. When k ≥ 3, we improve known partial results by removing the requirement of an invariant measure and related ergodicity assumptions. We are also for the first time able to work in low regularity.

Ruelle-Taylor resonances of Anosov actions

“…Higher rank R κ -Anosov actions have in particular been studied mostly for their rigidity: they are conjectured to be always smoothly conjugated to several models, mostly of algebraic nature (see e.g. the introduction of [SV19] for a precise statement and a state of the art on this question). The local rigidity of R κ -Anosov actions near standard Anosov actions 2 was proved in [KS94], and an important step of the proof relies on showing…”