$C^*$-algebras associated to non-homogeneous minimal systems and their K-theory

Abstract

Some background from topological dynamics and C Ã -algebrasAs a general reference on dynamical systems and C Ã -crossed products, we refer to the books by Walters [15] and Tomiyama [14].By a topological dynamical system, we mean a pair X Y 9, where X is a compact metric space and 9X X 3 X is a homeomorphism. To avoid trivialities we assume that X is infinite. We will be exclusively interested in the case where 9 is minimal. That is, if A is a closed subset of X and TA A, then A or A X . An equivalent formulation is that the orbit of x under 9, orb 9 x f9 n x X n P Zg, is dense in X for each x P X .

Crossed products by minimal homeomorphisms

“…k,i •h −(j−1) ) ⊂ h j (U k,i ) and the fact that the sets h j (U k,i ), for (k, i, j) ∈ Q, are disjoint. Part (11) is now immediate. In (12), the matrix unit relations are easy once one has the orthogonality relations in (10), and that E k,i is a finite dimensional C*-algebra with identity p k,i is then immediate.…”

“…The remaining examples involve disconnected spaces, so arguments using Exel's rotation numbers don't apply. The first is from Gjerde and Johansen [11], and is discussed in Example 1.6 of [33].…”

Non-homogeneous extensions of Cantor minimal systems

“…Haddad and Johnson in [8] showed that the set of such x could be much larger and even have positive measure under the unique invariant measure for (X, ϕ). More importantly for our purposes, Gjerde and Johansen [7] showed that the 3 ∞ -odometer could be replaced with any minimal system, (X, ϕ), with X a Cantor set. Their principal tool was the Bratteli-Vershik model for such systems [12,Chapters 4 and 5].…”

“…This example has been generalized in several ways (for example [1,2,7,8]). In Floyd's example, the points x with π −1 {x} infinite all lie in a single orbit.…”

Toeplitz flows, ordered Bratteli diagrams and strong orbit equivalence

“…The concept of 'strong orbit equivalence' plays a very important role in those connections and gives a new classification of dynamical systems (see [GJ2,Or,Su1,Su2,Su3]). …”