Asymptoticity, automorphism groups, and strong orbit equivalence

We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined K-theory). The technique developed in our study also enables us to compute the K-theory of certain amenable minimal Cantor systems. We apply it to the diagonal actions of the boundary actions and the products of the odometer transformations, and determine their K-theory. Then we classify them in terms of the topological full groups, continuous orbit equivalence, strong orbit equivalence, and the crossed products.

To any continuous eigenvalue of a Cantor minimal system $$(X,\,T)$$ , we associate an element of the dimension group $$K^0(X,\,T)$$ associated to $$(X,\,T)$$ . We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group $$(G,\,u)$$ (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

We associate different types of full groups to Cantor minimal systems. We show how these various groups (as abstract groups) are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy, respectively. Furthermore, we introduce a group homomorphism, the socalled mod map, from the normalizers of the various full groups to the automorphism groups of the (ordered)K 0-groups, which are associated to the Cantor minimal systems. We show how this in turn is related to the automorphisms of the associatedC *-crossed products. Our results are analogues in the topological dynamical setting of results obtained by Dye, Connes-Krieger and Hamachi-Osikawa in measurable dynamics.

Minimal homeomorphisms on the locally compact Cantor set are investigated. We prove that scaled dimension groups modulo infinitesimal subgroups determine topological orbit equivalence classes of locally compact Cantor minimal systems. We also introduce several full groups and show that they are complete invariants for orbit equivalence, strong orbit equivalence and flip conjugacy. These are locally compact versions of the famous results for Cantor minimal systems obtained by Giordano et al. Moreover, proper homomorphisms and skew product extensions of locally compact Cantor minimal systems are examined and it is shown that every finite group can be embedded into the group of centralizers trivially acting on the dimension group.

The algebraic invariants associated to the group actions on the Cantor set provide an interesting connection between the fields of dynamical systems and group theory. For instance, Giordano, Putnam and Skau have shown in [29] that the dimension group (see [24] for an introduction about dimension groups) of a minimal Z-action on the Cantor set completely determines its strong orbit equivalence class. Furthermore, the topological full group of such a system, which is known from Juschenko and Monod [38] to be amenable, determines its flip-conjugacy class (see [6] and [30] for more details). On the other hand, the amenability of the topological full groups of minimal Z-actions together with their properties shown in [41] by Matui make them the first known examples of infinite groups which are at the same time amenable, simple and finitely generated. Recently, another algebraic invariant, the group of automorphisms of actions on the Cantor set, has caught the eye of several researchers working in the field [13, 15, 16, 17, 14, 19, 20]. In [5], Boyle, Lind and Rudolph focused their attention on the group of automorphisms of subshifts of finite type, showing that these groups are always countable and residually finite. At the same time, they gave an example of a minimal Z-action on the Cantor set whose group of automorphisms contains Q, which implies that the automorphism group of a minimal action may be a non-residually finite group (recall that the Z-subshifts of finite type are not minimal). This leads to the natural question about the relation between the algebraic properties of the group of automorphisms and the dynamics of the system. Indeed, the residually finite property of the group of automorphisms of the subshifts of finite type is a consequence of the existence of periodic points.

Bounded orbit injection equivalence is an equivalence relation defined on minimal free Cantor systems which is a candidate to generalize flip Kakutani equivalence to actions of the Abelian free groups on more than one generator. This paper characterizes bounded orbit injection equivalence in terms of a mild strengthening of Rieffel-Morita equivalence of the associated C*-crossed-product algebras. Moreover, we construct an ordered group which is an invariant for bounded orbit injection equivalence, and does not agrees with the K0 group of the associated C*-crossed-product in general. This new invariant allows us to find sufficient conditions to strengthen bounded orbit injection equivalence to orbit equivalence and strong orbit equivalence.

To a Toeplitz flow$(X,T)$we associate an ordered$K^{0}$-group, denoted$K^{0}(X,T)$, which is order isomorphic to the$K^{0}$-group of the associated (non-commutative)$C^{\ast }$-crossed product$C(X)\rtimes _{T}\mathbb{Z}$. However,$K^{0}(X,T)$can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the$K^{0}$-groups that arise from Toeplitz flows$(X,T)$as exactly those simple dimension groups$(G,G^{+})$that contain a non-cyclic subgroup$H$of rank one that intersects$G^{+}$non-trivially. Furthermore, the Bratteli diagram realization of$(G,G^{+})$can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex$K$there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows$(X,T)$such that the set of$T$-invariant probability measures$M(X,T)$is affinely homeomorphic to$K$, where the entropy$h(T)$may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of$(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

We study certain countable locally finite groups attached to minimal homeomorphisms, and prove that the isomorphism relation on simple, countable, locally finite groups is a universal relation arising from a Borel $S_\infty $-action. This work also provid

Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable S {\mathcal S} -adic subshifts. This is done by establishing necessary and sufficient conditions for a minimal subshift to be of finite topological rank. As an application, we show that minimal subshifts with non-superlinear complexity (like many classical zero-entropy examples) have finite topological rank. Conversely, we analyze the complexity of S {\mathcal S} -adic subshifts and provide sufficient conditions for a finite topological rank subshift to have a non-superlinear complexity. This includes minimal Cantor systems given by Bratteli-Vershik representations whose tower levels have proportional heights and the so-called left to right S {\mathcal S} -adic subshifts. We also show that finite topological rank does not imply non-superlinear complexity. In the particular case of topological rank two subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial.

This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is arbitrarily close to a continuous contact or strictly contact dynamical system with the same end point. In particular, the above groups of time-one maps are independent of the choice of norm in the definition of the contact distance. On every contact manifold we construct topological contact dynamical systems with time-one maps that fail to be Lipschitz continuous, and smooth contact vector fields whose flows are topologically conjugate but not conjugate by a contact C^1-diffeomorphism.

In Chapter 10 we showed how a topological dynamical system \( (K;\varphi) \) gives rise to a measure-preserving system by choosing a \( \varphi \)-invariant measure on K. The existence of such a measure is guaranteed by the Krylov–Bogoljubov Theorem 10.2. In general there can be many invariant measures, and we also investigated how minimality of the topological system is reflected in properties of the associated measure-preserving system. It is now our aim to go in the other direction: starting from a given measure-preserving system we shall construct some topological system (sometimes called topological model) and an invariant measure so that the resulting measure-preserving system is isomorphic to the original one. By doing this, methods from the theory of topological dynamical system will be available, and we may gain further insights into measure-preserving systems. Thus, by switching back and forth between the measure theoretic and the topological situation, we can deepen our understanding of dynamical systems. In particular, this procedure will be carried out in Chapter 17

The existence of semi-horseshoes for partially hyperbolic diffeomorphisms

On automorphism groups of Toeplitz subshifts, Discrete Analysis 2017:11, 19 pp. A discrete dynamical system is a space $X$ with some kind of structure, together with a map $\sigma\colon X\to X$ that preserves the structure. (For instance, if $X$ is a topological space, then one asks for $\sigma$ to be continuous, and if $X$ is a differentiable manifold, then one asks for it to be a diffeomorphism.) Given such a system, one studies the structure of the orbits $x, \sigma x, \sigma^2x, \dots$ that are obtained by iterating the map $\sigma$. A particularly interesting subfield of dynamics is _symbolic dynamics_, where $X$ is a space of bi-infinite sequences over a finite alphabet $A$, $X$ is closed under the left shift, and $\sigma$ is that left shift. One also asks for $X$ to be closed in the topological sense: we take the discrete topology on $A$ and the product topology on $A^{\mathbb Z}$, of which $X$ is a subset. A system $(X,\sigma)$ is called a _shift space_. Such spaces can encode interesting combinatorial information, and that has led to a very fruitful interplay between combinatorics and dynamical systems. An _automorphism_ of the system $(X, \sigma)$ is a homeomorphism $\phi\colon X\to X$ that commutes with $\sigma$, and ${\rm Aut}(X, \sigma)$ denotes the group (under composition) of automorphisms of the system. The _complexity_ of a shift system ${\rm Aut}(X, \sigma)$ is the map $p\colon\mathbb N\to \mathbb N$ that counts the number of blocks of length $n$ appearing in the sequences $x\in X$. If the complexity is linear, then the automorphism group is understood for any shift ${\rm Aut}(X, \sigma)$, but beyond linear, the problem becomes complicated. For example, under mild assumptions on the shift ${\rm Aut}(X, \sigma)$, the automorphism group is not finitely generated and it contains isomorphic copies of all finite groups, countably many copies of $\mathbb Z$, and the free group on any finite number of generators. Thus while ${\rm Aut}(X, \sigma)$ is always countable, in general it can be quite complicated and difficult to compute. However, for several reasons it is desirable to do so: for example, it gives a useful invariant. This paper continues recent work on automorphism groups for various classes of shift spaces, computing the automorphism group for the class of Toeplitz shifts, a large class of shift systems frequently used to provide counterexamples in symbolic dynamics. A sequence $x\in A^{\mathbb Z}$ is _Toeplitz_ if every finite block in $x$ appears periodically, and a shift space $(X, \sigma)$ is a _Toeplitz shift_ if $X$ is the orbit closure of some Toeplitz sequence. (It is not hard to construct Toeplitz sequences that are not periodic. For one example, take $x_n$ to be the parity of $k$, where $k$ is maximal such that $2^k|n$.) This rigid structure on $X$ implies that ${\rm Aut}(X, \sigma)$ is Abelian, and this is the starting point for the classification of the automorphism groups of Toeplitz shifts. The authors start with Toeplitz shifts of subquadratic complexity, showing that the automorphism group is spanned by the roots of the shift map $\sigma$ modulo the torsion subgroup $T$ of ${\rm Aut}(X, \sigma)$. More generally, they show that if ${\rm Aut}(X, \sigma)/\langle\sigma\rangle$ is a periodic group, then the automorphism group is spanned by $T$ and the roots of the shift $\sigma$ (that is, the automorphisms $\phi$ such that $\phi^n=\sigma$ for some $n$). Under the further assumption that $T$ is trivial, they show that the automorphism group is either infinite cyclic or is not finitely generated. This method leads to examples of Toeplitz shifts whose complexity is arbitrarily close to linear, in the sense that for every $\varepsilon > 0$ the complexity satisfies the inequality $p(n)\leq Cn^{1+\varepsilon}$ for some constant $C=C_\varepsilon > 0$, such that the automorphism group is not finitely generated. Note that this result cannot be extended to linear complexity, where it is known that the automorphism group is always finitely generated. In the opposite regime, that of high complexity, the authors show that the automorphism group need not be large. Given any infinite and finitely generated Abelian group $G$ with cyclic torsion, they construct a Toeplitz shift with positive entropy (meaning that the complexity function grows exponentially) whose automorphism group is exactly $G$.

Given a topological dynamical system (X, T) and an arithmetic function u: ℕ → ℂ, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T). It is proved that, given an ergodic measure-preserving system (Z, $$\mathcal{D}$$ , к, R),the strong MOMO propertly (relately to u) of a uniquely ergodic midel (X, T)of R yields all other uniquely ergodic midel of R to be u-disjiont. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue—Viorse and Rudin—Shapiro substitutions), systems determined by Kakutani sequences are Mobius (and Liouville) disjoint. The validity of Sarnak5s conjecture implies the strong MOMO property relatively to μ in all zero entropy systems; in particular, it makes μ-disjointness uniform. The absence of the strong MOMO property in positive entropy systems is discussed and it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.