The symmetries of affine $K$-systems and a program for centralizer rigidity
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2025
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| author | Damjanović, Danijela Wilkinson, Amie Wu, Chengyang Xu, Disheng |
| author_facet | Damjanović, Danijela Wilkinson, Amie Wu, Chengyang Xu, Disheng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The symmetries of affine $K$-systems and a program for centralizer rigidity Damjanović, Danijela Wilkinson, Amie Wu, Chengyang Xu, Disheng Dynamical Systems 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. |
| title | The symmetries of affine $K$-systems and a program for centralizer rigidity |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.09084 |