The symmetries of affine $K$-systems and a program for centralizer rigidity

Abstract

Let Aff(X) be the group of affine diffeomorphisms of a closed homogeneous manifold X=G/B admitting a G-invariant Lebesgue-Haar probability measure $\mu$. For $f_0\in$ Aff(X), let $Z^{\infty }(f_0)$ be the group of $C^{\infty }$ diffeomorphisms of X commuting with $f_0$. This paper addresses the question: for which $f_0\in$ Aff(X) is $Z^{\infty }(f_0)$ a Lie subgroup of $Dif{f}^{\infty }(X)$? Among our main results are the following. (1) If $f_0\in$ Aff(X) is weakly mixing with respect to $\mu$, then $Z^{\infty }(f_0)<$ Aff(X), and hence is a Lie group. (2) If $f_0\in$ Aff(X) is ergodic with respect to $\mu$, then $Z^{\infty }(f_0)$ is a (necessarily $C^0$ closed) Lie subgroup of $Dif{f}^{\infty }(X)$ (although not necessarily a subgroup of Aff(X)). (3) If $f_0\in$ Aff(X) fails to be a K-system with respect to $\mu$, then there exists $f\in$ Aff(X) arbitrarily close to $f_0$ such that $Z^{\infty }(f)$ is not a Lie group, containing as a continuously embedded subgroup either the abelian group $C_c^{\infty }((0,1))$ (under addition) or the simple group $Dif{f}_c^{\infty }((0,1))$ (under composition). (4) Considering perturbations of $f_0$ by left translations, we conclude that $f_0$ is stably ergodic if and only if the condition $Z^{\infty }<$ Aff(X) holds in a neighborhood of $f_0$ in Aff(X). (Note that by BS97, Dani77, $f_0\in$ Aff(X) is stably ergodic in Aff(X) if and only if $f_0$ is a K-system.) The affine K-systems are precisely those that are partially hyperbolic and essentially accessible, belonging to a class of diffeomorphisms whose dynamics have been extensively studied. In addition, the properties of partial hyperbolicity and accessibility are stable under $C^1$-small perturbation, and in some contexts, essential accessibility has been shown to be stable under smooth perturbation. Considering the smooth perturbations of affine K-systems, we outline a full program for (local) centralizer rigidity.