Global smooth rigidity for toral automorphisms

TL;DR

This paper proves that conjugacies between certain toral automorphisms and smooth diffeomorphisms are infinitely differentiable under weak irreducibility conditions, enhancing previous rigidity results.

Contribution

It establishes $C^ abla$ regularity of conjugacies for weakly irreducible hyperbolic and partially hyperbolic automorphisms, improving prior rigidity theorems.

Findings

$C^ abla$ conjugacy regularity for weakly irreducible hyperbolic automorphisms

$C^ abla$ conjugacy regularity for weakly irreducible partially hyperbolic automorphisms

Improved regularity results in local and global rigidity contexts

Abstract

We study regularity of a conjugacy between a hyperbolic or partially hyperbolic toral automorphism $L$ and a $C^\infty$ diffeomorphism $f$ of the torus. For a very weakly irreducible hyperbolic automorphism $L$ we show that any $C^1$ conjugacy is $C^\infty$. For a very weakly irreducible ergodic partially hyperbolic automorphism $L$ we show that any $C^{1+\text{H\"older}}$ conjugacy is $C^\infty$. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local and global rigidity results.