The symmetries of affine $K$-systems and a program for centralizer rigidity
Danijela Damjanović, Amie Wilkinson, Chengyang Wu, Disheng Xu
TL;DR
The paper investigates the symmetry groups of affine diffeomorphisms on compact homogeneous spaces X=G/B, focusing on when the smooth centralizer ${\mathcal{Z}}^\infty(f)$ is a Lie group. It establishes a sharp link between ergodic properties (ergodic, weakly mixing, $K$-systems) and centralizer structure: affine $K$-systems are precisely those for which perturbations retain a finite-dimensional Lie centralizer, and in many cases the interior of the set of affine maps with affine centralizers coincides with ergodic and $K$-system interiors. A perturbative program is proposed to study centralizer rigidity under affine and smooth perturbations, identifying rank-1 factors as the main obstruction to rigidity and outlining a path to local centralizer rigidity in terms of partial hyperbolicity and essential accessibility. The authors develop a detailed framework using towers of homogeneous fiber bundles and special bundle morphisms to prove that non-$K$ affine maps can have centralizers containing large non-Lie subgroups, while affine $K$-systems exhibit robust Lie centralizers under perturbations. The results unify previous rigidity phenomena in affine settings and lay out conjectures and methods for broader centralizer rigidity in smooth dynamics. Overall, the work advances a program linking ergodic theory, Lie theory, and perturbative rigidity to illuminate the algebraic structure of symmetries in smooth dynamical systems. All results are carefully embedded in the language of affine actions on homogeneous spaces and their perturbations, with a clear pathway toward a comprehensive local rigidity theory for centralizers.
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 $μ$. 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 $Diff^\infty(X)$? Among our main results are the following. (1) If $f_0\in$ Aff(X) is weakly mixing with respect to $μ$, 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 $μ$, then $Z^\infty(f_0)$ is a (necessarily $C^0$ closed) Lie subgroup of $Diff^\infty(X)$ (although not necessarily a subgroup of Aff(X)). (3) If $f_0\in$ Aff(X) fails to be a K-system with respect to $μ$, 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 $C^\infty_c((0,1))$ (under addition) or the simple group $Diff^\infty_c((0,1))$ (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.