Minimal Homeomorphisms Research Articles

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.

Rokhlin Dimension and C*-Dynamics

We develop the concept of Rokhlin dimension for integer and for finite group actions on C*-algebras. Our notion generalizes the so-called Rokhlin property, which can be thought of as Rokhlin dimension 0. We show that finite Rokhlin dimension is prevalent and appears in cases in which the Rokhlin property cannot be expected: the property of having finite Rokhlin dimension is generic for automorphisms of $$\mathcal{Z}$$ -stable C*-algebras, where $$\mathcal{Z}$$ denotes the Jiang–Su algebra. Moreover, crossed products by automorphisms with finite Rokhlin dimension preserve the property of having finite nuclear dimension, and under a mild additional hypothesis also preserve $${\mathcal{Z}}$$ -stability. In topological dynamics our notion may be interpreted as a topological version of the classical Rokhlin lemma: automorphisms arising from minimal homeomorphisms of finite dimensional compact metrizable spaces always have finite Rokhlin dimension. The latter result has by now been generalized by Szabó to the case of free and aperiodic $${\mathbb{Z}^{d}}$$ -actions on compact metrizable and finite dimensional spaces.

Full groups of minimal homeomorphisms and Baire category methods

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

Measures on Cantor sets: The good, the ugly, the bad

We translate Akin’s notion of good (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups). This yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of refinability, we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions are also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results.

Regularity of topological cocycles of a class of non-isometric minimal homeomorphisms

We study topological cocycles of a class of non-isometric distal minimal homeomorphisms of multidimensional tori, introduced by Furstenberg in [5] as iterated skew product extensions by the torus, starting with an irrational rotation.We prove that there are no topological type ${III}_0$ cocycles of these homeomorphisms with values in an Abelian locally compact group.Moreover, under the assumption that the Abelian locally compact group has no non-trivial connected compact subgroup, we show that a topologically recurrent cocycle is always regular, i.e. it is topologically cohomologous to a cocycle with values only in the essential range.These properties are well-known for topological cocycles of minimal rotations on compact metric groups (cf. [6], [2], [9], and [10]), but the distal minimal homeomorphisms considered in this paper are far from the isometric behaviour of minimal rotations and do not admit rigidity times.

On the dynamics of nonreducible cylindrical vortices

We study the dynamics of Euclidean isometric extensions of minimal homeomorphisms of compact metric spaces. Under a general hypothesis of homogeneity for the base space, we show that these systems are never minimal, thus extending a classical result of Besicovitch concerning cylindrical cascades. Moreover, using Anosov–Katok-type methods, we construct a topologically transitive isometric extension over an irrational rotation with a two-dimensional fiber.

Minimal non-invertible transformations of solenoids

We construct a continuous non-invertible minimal transformation of an arbitrary solenoid. Since solenoids, as all other compact monothetic groups, also admit minimal homeomorphisms, our result allows one to classify solenoids among continua admitting both

Nilpotent extensions of Furstenberg transformations

We study topologically recurrent cocycles of a class of minimal homeomorphisms on tori, usually called Furstenberg transformations, and with values in nilpotent locally compact second countable groups. We prove that these cocycles fulfil a regularity property and admit a partitioning of the skew product into orbit closures.

Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett–Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.

A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory

We use ergodic theory to prove a quantitative version of a theorem of M.A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a structure theorem for continuous matrix cocycles over minimal homeomorphisms having the property that all forward products are uniformly bounded.

Crossed products by minimal homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h : X → X be a minimal homeomorphism. We show that the associated crossed product C*-algebra A = C*(ℤ, X, h) has tracial rank zero whenever the image of K0(A) in Aff(T(A)) is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values.

Minimal homeomorphisms on low-dimensional tori

AbstractWe study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

Tracial Rokhlin property for automorphisms on simple -algebras

AbstractLet A be a unital simple $A\mathbb {T}$-algebra of real rank zero. Given an isomorphismγ1:K1(A)→K1(A), we show that there is an automorphism α:A→A such that α*1=γ1 and α has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes _{\alpha }\mathbb {Z}$ is a simple unital AH-algebra with real rank zero. We also show that automorphisms with the Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.

Minimal homeomorphisms and approximate conjugacy in measure

Let $X$ be an infinite compact metric space with finite covering dimension. Let $\af,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed products are dense in the space of real affine continuous functions on the tracial state space. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.

Recursive subhomogeneous algebras

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form C ( X , M n ) C (X, M_n) many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if A A is a separable C*-algebra whose irreducible representations all have dimension at most N > ∞ , N > \infty , and if for each n n the space of n n -dimensional irreducible representations has finite covering dimension, then A A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.

On the subshift within a strong orbit equivalence class for minimal homeomorphisms

In this paper we will show that, for any strong orbit equivalence class for Cantor minimal systems, the class contains (minimal) subshifts which attain arbitrary finite topological entropy.

Classification of homomorphisms and dynamical systems

Let A be a unital simple C*-algebra, with tracial rank zero and let X be a compact metric space. Suppose that h 1 ,h 2 : C(X) → A are two unital monomorphisms. We show that h 1 and h 2 are approximately unitarily equivalent if and only if [h 1 ] = [h 2 ] in KL(C(X), A) and τ o h 1 (f) = τ o h 2 (f) for every f ∈ C(X) and every trace r of A. Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and let α, β: X → X be two minimal homeomorphisms. Using the above-mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with the strong orbit equivalence of Giordano, Putnam and Skau, and the K-theoretical condition is equivalent to saying that the associate crossed product C*-algebras are isomorphic. Another application of the above-mentioned result is given for C*-dynamical systems related to a problem of Kishimoto. Let A be a unital simple AH-algebra with no dimension growth and with real rank zero, and let a ∈ Aut(A). We prove that if α r fixes a large subgroup of K 0 (A) and has the tracial Rokhlin property, then Ax α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.

Nilpotent extensions of minimal homeomorphisms

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well-known topological essential range if the group is abelian. We then define a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore, we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

The Relationship Between Entropy and Strong Orbit Equivalence for the Minimal Homeomorphisms (I)

For every minimal homeomorphism of a Cantor set its strongly orbit equivalence class contains homeomorphisms of all possible finite entropies.

Topological orbit equivalence of locally compact Cantor minimal systems

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.