On classification of higher rank Anosov actions on compact manifold

TL;DR

This paper establishes smooth classification results for higher-rank Anosov actions on compact manifolds, introducing new methods that relax previous restrictions and apply broadly to dynamical systems.

Contribution

It proves a new standard form for derivative cocycles of TNS totally Anosov Z^k actions without previous constraints, using a novel non-uniform redefining argument.

Findings

Established smooth classification under joint integrability, resonance-free, or Lyapunov pinching conditions.

Introduced a new standard form for derivative cocycles of Anosov actions.

Developed a non-uniform redefining argument for continuity of dynamical objects.

Abstract

We prove global smooth classification results for TNS totally Anosov Z^k actions on general compact manifolds, under each one of the following conditions: joint integrability, resonance-free or Lyapunov pinching condition. Unlike the previous results, we do not require any uniform quasiconformality or pinching condition of action elements on coarse Lyapunov distributions, nor do we have any restriction on the dimension of coarse Lyapunov distributions. The main novelty is in proving a new standard form of the derivative cocycle for any TNS totally Anosov Z^k action on general manifold. A main idea is to create a new mechanism called a non-uniform redefining argument to prove continuity of general dynamically-defined object, which should apply to more general rigidity problems in dynamical systems.