Dimension Groups of Topological Joinings and Non-coalescence of Cantor Minimal Systems

نویسندگان

  • Hiroki Matui
  • HIROKI MATUI
چکیده

By a topological dynamical system (Y, ψ), we mean a compact Hausdorff space Y endowed with a homeomorphism ψ. When (Yi, ψi), i = 0, 1 are two topological dynamical systems, ψ0 × ψ1-invariant closed subsets of Y0 × Y1 are called (topological) joinings, and when (Y0, ψ0) equals (Y1, ψ1), they are called self-joinings. In the measure-theoretical setting, the notion of selfjoinings was introduced by D. Rudolph in [R], and it was proved that the minimal self-joining property implies coalescence and zero entropy. In this paper, we will compute the dimension group of joinings of Cantor minimal systems. When Y is the Cantor set and a homeomorphism ψ on Y has no nontrivial invariant closed set, (Y, ψ) is called a Cantor minimal system. We define the dimension group K0(Y, ψ) of (Y, ψ) as the quotient of C(Y,Z) by the coboundary subgroup Bψ = {f − f ◦ ψ−1; f ∈ C(Y,Z)}. In [GPS], it was proved that the dimension group K0(Y, ψ), as an ordered group with a distinguished order unit, characterizes the strong orbit equivalence class of (Y, ψ). We would like to consider the case that a joining (Z, τ) of Cantor minimal systems (Y0, ψ0) and (Y1, ψ1) is also a Cantor minimal system. (We must distinguish the property of minimal self-joinings and minimal systems in the joinings.) We don’t know a necessary and sufficient condition so that the joining is minimal. In a special case, however, we will prove that the joining becomes a Cantor minimal system, and compute its dimension group. Our

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تاریخ انتشار 2002