اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn continuous orbit equivalence rigidity for virtually cyclic group actions
83
0
0.0
(
0
)
تأليف
Yongle Jiang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
قيم البحث
اقرأ أيضاً
For groups of diffeomorphisms of $T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $T^2$ up to both topological conjugacy and smooth conjugacy under mild assumptions. Along
the way, we also prove a Tits alternative type theorem for some groups of diffeomorphisms of $T^2$.
By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variat
ions of) these groups are invariants for topologically free actions under continuous orbit equivalence.
In this paper, we study a natural class of groups that act as affine transformations of $mathbb T^N$. We investigate whether these solvable, abelian-by-cyclic, groups can act smoothly and nonaffinely on $mathbb T^N$ while remaining homotopic to the a
ffine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $C^r$ diffeomorphims can be conjugated to the affine action by $C^{r-epsilon}$ conjugacy. Next, we show that in any dimension, $C^1$ small perturbations can be conjugated to an affine action via $C^{1+epsilon}$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.
We determine the Krieger type of nonsingular Bernoulli actions $G curvearrowright prod_{g in G} ({0,1},mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $mu_g$. We prove in particular that the action is never of type II$_infty$
if $G$ is abelian and not locally finite, answering Krengels question for $G = mathbb{Z}$. When $G$ is locally finite, we prove that type II$_infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.
We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally fin
ite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
