Minimal Dynamical Systems Research Articles

A geometric Elliott invariant and noncommutative rigidity of mapping tori

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C⁎-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C⁎-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

Application of waist inequality to entropy and mean dimension

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps π : ( X , T ) → ( Y , S ) \pi : (X, T) \to (Y, S) between dynamical systems and assume that the mean dimension of the domain ( X , T ) (X, T) is larger than the mean dimension of the target ( Y , S ) (Y, S) . We exhibit several situations for which the maps π \pi necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss [Israel J. Math. 115 (2000), pp. 1–24] about minimal dynamical systems non-embeddable in [ 0 , 1 ] Z [0,1]^{\mathbb {Z}} .

A𝕋-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call “fiberwise essentially minimal”, that is a class that includes essentially minimal systems and systems in which every orbit is minimal. We prove that the associated crossed product [Formula: see text]-algebra of such a system is an A[Formula: see text]-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product [Formula: see text]-algebra has real rank zero, which tells us that such [Formula: see text]-algebras are classifiable by [Formula: see text]-theory. The associated crossed product [Formula: see text]-algebras to these nontrivial examples are of particular interest because they are non-simple (unlike in the minimal case).

On the Structure Theory of Cubespace Fibrations

We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $$f :X\rightarrow Y$$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over Y. If the structure groups of f are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of f are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relative to factor maps between minimal topological dynamical systems for an arbitrary acting group. We prove that any factor map between minimal distal systems is a fibration and conclude that if such a map is of finite degree then it factors as a (possibly countable) tower of principal abelian Lie compact group extensions, thus achieving a refinement of both the Furstenberg’s and the Bronstein–Ellis structure theorems in this setting.

Multi-transitivity in non-autonomous discrete systems

We introduce and study the notions of multi-transitivity and thick transitivity for non-autonomous discrete dynamical systems. We obtain a sufficient condition under which in minimal non-autonomous discrete dynamical systems, multi-transitivity, thick transitivity and weakly mixing of all orders are equivalent. Counterexamples are constructed justifying that certain results related to above concepts which are true in autonomous discrete dynamical systems need not be true in non-autonomous discrete dynamical systems.

Embedding minimal dynamical systems into Hilbert cubes

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$. This problem is intimately related to the theory of mean dimension, which counts the average number of parameters for describing a dynamical system. Lindenstrauss proved that minimal systems of mean dimension less than cN for $$c=1/36$$ can be embedded in $$\left( [0,1]^N\right) ^{\mathbb {Z}}$$, and asked what is the optimal value for c. We solve this problem by showing embedding is possible when $$c=1/2$$. The value $$c=1/2$$ is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques.

Minimal dynamical systems model of the Northern Hemisphere jet stream via embedding of climate data

Abstract. We derive a minimal dynamical systems model for the Northern Hemisphere midlatitude jet dynamics by embedding atmospheric data and by investigating its properties (bifurcation structure, stability, local dimensions) for different atmospheric flow regimes. The derivation is a three-step process: first, we obtain a 1-D description of the midlatitude jet stream by computing the position of the jet at each longitude using ERA-Interim. Next, we use the embedding procedure to derive a map of the local jet position dynamics. Finally, we introduce the coupling and stochastic effects deriving from both atmospheric turbulence and topographic disturbances to the jet. We then analyze the dynamical properties of the model in different regimes: one that gives the closest representation of the properties extracted from real data; one featuring a stronger jet (strong coupling); one featuring a weaker jet (weak coupling); and one with modified topography. Our model, notwithstanding its simplicity, provides an instructive description of the dynamical properties of the atmospheric jet.

The nonexistence of universal metric flows

We consider dynamical systems of the form $(X,f)$ where $X$ is a compact metric space and $f\colon X\to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $\omega$-limit sets, answering a question by Will Brian.

Multiplicative combinatorial properties of return time sets in minimal dynamical systems

We investigate the relationship between the dynamical properties of minimal topological dynamical systems and the multiplicative combinatorial properties of return time sets arising from those systems. In particular, we prove that for a residual sets of points in any minimal system, the set of return times to any non-empty, open set contains arbitrarily long geometric progressions. Under the separate assumptions of total minimality and distality, we prove that return time sets have positive multiplicative upper Banach density along $\mathbb{N}$ and along multiplicative subsemigroups of $\mathbb{N}$, respectively. The primary motivation for this work is the long-standing open question of whether or not syndetic subsets of the positive integers contain arbitrarily long geometric progressions; our main result is some evidence for an affirmative answer to this question.

Crossed products and minimal dynamical systems

Let [Formula: see text] be an infinite compact metric space with finite covering dimension and let [Formula: see text] be two minimal homeomorphisms. We prove that the crossed product [Formula: see text]-algebras [Formula: see text] and [Formula: see text] are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if [Formula: see text] is an infinite compact metric space and if [Formula: see text] is a minimal homeomorphism such that [Formula: see text] has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This implies that the crossed product is in a classifiable class of amenable simple [Formula: see text]-algebras.

Rank 2 proximal Cantor systems are residually scrambled

ABSTRACTDownarowicz and Maass [7] proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of the essentially minimal zero-dimensional systems. We consider the cases in which topological ranks are 2 and unique minimal sets are fixed points. Akin and Kolyada [2], had shown that if the unique minimal set of an essentially minimal system is a fixed point, then the system must be proximal. The finite topological rank implies expansiveness; furthermore, in the case of proximal Cantor systems with topological rank 2, the expansiveness is always from the lowest degree. Rank 2 proximal Cantor systems are residually scrambled. We present a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of the systems that are topologically mixing can be 1 and 2. Moreover, we present examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of the systems that are not weakly mixing can be 1 and 2.

The structure of tame minimal dynamical systems for general groups

We use the structure theory of minimal dynamical systems to show that, for a general group $\Gamma$, a tame, metric, minimal dynamical system $(X, \Gamma)$ has the following structure: \begin{equation*} \xymatrix {& \tilde{X} \ar[dd]_\pi \ar[dl]_\eta & X^* \ar[l]_-{\theta^*} \ar[d]^{\iota} \ar@/^2pc/@{>}^{\pi^*}[dd]\\ X & & Z \ar[d]^\sigma\\ & Y & Y^* \ar[l]^\theta } \end{equation*} Here (i) $\tilde{X}$ is a metric minimal and tame system (ii) $\eta$ is a strongly proximal extension, (iii) $Y$ is a strongly proximal system, (iv) $\pi$ is a point distal and RIM extension with unique section, (v) $\theta$, $\theta^*$ and $\iota$ are almost one-to-one extensions, and (vi) $\sigma$ is an isometric extension. When the map $\pi$ is also open this diagram reduces to \begin{equation*} \xymatrix {& \tilde{X} \ar[dl]_\eta \ar[d]^{\iota} \ar@/^2pc/@{>}^\pi[dd]\\ X & Z \ar[d]^\sigma\\ & Y } \end{equation*} In general the presence of the strongly proximal extension $\eta$ is unavoidable. If the system $(X, \Gamma)$ admits an invariant measure $\mu$ then $Y$ is trivial and $X = \tilde{X}$ is an almost automorphic system; i.e. $X \overset{\iota}{\to} Z$, where $\iota$ is an almost one-to-one extension and $Z$ is equicontinuous. Moreover, $\mu$ is unique and $\iota$ is a measure theoretical isomorphism $\iota : (X,\mu, \Gamma) \to (Z, \lambda, \Gamma)$, with $\lambda$ the Haar measure on $Z$. Thus, this is always the case when $\Gamma$ is amenable.

Construction of minimal skew products of amenable minimal dynamical systems

For an amenable minimal topologically free dynamical system \alpha of a group on a compact metrizable space Z and for a compact metrizable space Y satisfying a mild condition, we construct a minimal skew product extension of \alpha on Z \times Y . This generalizes a result of Glasner and Weiss. We also study the pure infiniteness of the crossed products of minimal dynamical systems arising from this result. In particular, we give a generalization of a result of Rørdam and Sierakowski.

On the spectrum of operator families on discrete groups over minimal dynamical systems

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $$\ell ^p ({\mathbb {Z}})$$ . Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $$\ell ^p$$ -spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.

Constancy of spectra of equivariant (non‐selfadjoint) operators over minimal dynamical systems

AbstractWe show that for a large class of equivariant continuous, possibly non‐selfadjoint, operators over minimal dynamical systems the spectrum as a set is in fact independent of the random parameter. Furthermore, the spectrum agrees with the essential spectrum. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Minimal Dynamical Systems on Connected Odd Dimensional Spaces

AbstractLetbe a minimal homeomorphism (n ≥1). We show that the crossed producthas rational tracial rank at most one. Let Ω be a connected, compact, metric space with finite covering dimension and with. Suppose that,where Giis a finite abelian group, i = 0,1. Let β:Ω→Ωbe a minimal homeomorphism. We also show thathas rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on X✗Ω, where X is the Cantor set.

Formula omitted]-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β:Sn→Sn, for n=2k+1, k≥1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (Sn,β) there is a Cantor minimal system (X,α) such that the corresponding product system (X×Sn,α×β) is minimal and the resulting crossed product C⁎-algebra C(X×Sn)⋊α×βZ is tracially approximately an interval algebra (TAI). This entails classification for such C⁎-algebras. Moreover, the minimal Cantor system (X,α) is such that each tracial state on C(X×Sn)⋊α×βZ induces the same state on the K0-group and such that the embedding of C(Sn)⋊βZ into C(X×Sn)⋊α×βZ preserves the tracial state space. This implies C(Sn)⋊βZ is TAI after tensoring with the universal UHF algebra, which in turn shows that the C⁎-algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.

UHF-slicing and classification of nuclear C*-algebras

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.

Crossed product [formula omitted]-algebras of minimal dynamical systems on the product of the Cantor set and the torus

This paper studies the relationship between minimal dynamical systems on the product of the Cantor set X and the torus T2 and their corresponding crossed product C⁎-algebras. When the cocycles are rotations, it is shown that the crossed product C⁎-algebras have tracial rank no more than one, thus these C⁎-algebras are classifiable by the Elliott invariant. For two such systems, while assuming certain rigidity condition on traces, we prove that if there is certain isomorphism between the ordered K0 of the two crossed product C⁎-algebras, then these two systems are approximately K-conjugate. Our proof also indicates that C⁎-strongly flip conjugacy implies approximate K-conjugacy in this case.