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Register a new userGlobal Rigidity of Some Abelian-by-Cyclic group actions on $T^2$
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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 $T^2$.
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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 affine 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 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.
In this work, we investigate the dynamical and geometric properties of weak solenoids, as part of the development of a calculus of group chains associated to Cantor minimal actions. The study of the properties of group chains was initiated in the works of McCord 1965 and Fokkink and Oversteegen 2002, to study the problem of determining which weak solenoids are homogeneous continua. We develop an alternative condition for the homogeneity in terms of the Ellis semigroup of the action, then investigate the relationship between non-homogeneity of a weak solenoid and its discriminant invariant, which we introduce in this work. A key part of our study is the construction of new examples that illustrate various subtle properties of group chains that correspond to geometric properties of non-homogeneous weak solenoids.
The purpose of this paper is to study the phenomenon of large intersections in the framework of multiple recurrence for measure-preserving actions of countable abelian groups. Among other things, we show: (1) If $G$ is a countable abelian group and $varphi, psi : G to G$ are homomorphisms such that $varphi(G)$, $psi(G)$, and $(psi - varphi)(G)$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{varphi(g)}^{-1}A cap T_{psi(g)}^{-1}A) > mu(A)^3 - varepsilon}$ is syndetic. (2) If $G$ is a countable abelian group and $r,s in mathbb{Z}$ are integers such that $rG$, $sG$, and $(r pm s)G$ have finite index in $G$, then for every ergodic measure-preserving system $(X, mathcal{B}, mu, (T_g)_{g in G})$, every set $A in mathcal{B}$, and every $varepsilon > 0$, the set ${g in G : mu(A cap T_{rg}^{-1}A cap T_{sg}^{-1}A cap T_{(r+s)g}^{-1}A) > mu(A)^4 - varepsilon}$ is syndetic. In particular, these extend and generalize results of Bergelson, Host, and Kra concerning $mathbb{Z}$-actions and of Bergelson, Tao, and Ziegler concerning $mathbb{F}_p^{infty}$-actions. Using an ergodic version of the Furstenberg correspondence principle, we obtain new combinatorial applications. We also discuss numerous examples shedding light on the necessity of the various hypotheses above. Our results lead to a number of interesting questions and conjectures, formulated in the introduction and at the end of the paper.
Let $BS(1,n) =< a, b | aba^{-1} = b^n >$ be the solvable Baumslag-Solitar group, where $ ngeq 2$. It is known that BS(1,n) is isomorphic to the group generated by the two affine maps of the real line: $f_0(x) = x + 1$ and $h_0(x) = nx $. This paper deals with the dynamics of actions of BS(1,n) on closed orientable surfaces. We exhibit a smooth BS(1,n) action without finite orbits on $TT ^2$, we study the dynamical behavior of it and of its $C^1$-pertubations and we prove that it is not locally rigid. We develop a general dynamical study for faithful topological BS(1,n)-actions on closed surfaces $S$. We prove that such actions $<f,h | h circ f circ h^{-1} = f^n>$ admit a minimal set included in $fix(f)$, the set of fixed points of $f$, provided that $fix(f)$ is not empty. When $S= TT^2$, we show that there exists a positive integer $N$, such that $fix(f^N)$ is non-empty and contains a minimal set of the action. As a corollary, we get that there are no minimal faithful topological actions of BS(1,n) on $TT^2$. When the surface $S$ has genus at least 2, is closed and orientable, and $f$ is isotopic to identity, then $fix(f)$ is non empty and contains a minimal set of the action. Moreover if the action is $C^1$ then $fix(f)$ contains any minimal set.
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