On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

Let S denote the unit circle on the complex plane and ★:S2→S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,★) is isomorphic to the group (S,·); thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S,·) into (S,★) and obtain, in this way, the following description of operation ★: x★y=F(F−1(x)·F−1(y)) for x,y∈S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.

Decomposition rank of approximately subhomogeneous C*-algebras

It is shown that every Jiang–Su stable approximately subhomogeneous {{\mathrm{C}^{*}}}-algebra has finite decomposition rank. This settles a key direction of the Toms–Winter conjecture for simple approximately subhomogeneous {{\mathrm{C}^{*}}}-algebras. A key step in the proof is that subhomogeneous {{\mathrm{C}^{*}}}-algebras are locally approximated by a certain class of more tractable subhomogeneous algebras, namely a non-commutative generalization of the class of cell complexes. The result is applied, in combination with other recent results, to show classifiability of crossed product {{\mathrm{C}^{*}}}-algebras associated to minimal homeomorphisms with mean dimension zero.

Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid {\mathrm{C}^{*}} -algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

Pointwise Topological Stability

We decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a topologically stable point of a homeomorphism f as a point x such that for any C0-perturbation g of f there is a continuous semiconjugation defined on the g-orbit closure of x which tends to the identity as g tends to f. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.

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.

Dynamical simplices and Fraïssé theory

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$ . We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.