Ergodic Theory via Joinings

Independence in topological and C*-dynamics

“…Employing a local variational principle, Glasner showed in [30,Theorem 4. (3)] (see also [31,Theorem 19.27]) that for Z-systems the set of proximal entropy pairs is dense in the set of entropy pairs. As a consequence of Theorem 3.18 we have the following improvement:…”

“…To conclude Section 3 we discuss how the theory readily extends to actions of discrete amenable groups. Note in contrast that the measure-dynamical approach as it has been developed for Z-systems does not directly extend to the general amenable case, as one needs for example to find a substitute for the procedure of taking powers of a single automorphism (see [43] and Section 19.3 of [31]). …”

On tame dynamical systems

“…As was mentioned in the Introduction the following theorem is due to Köhler; our proof, though, is different (see also [22,Lemma 1.49]).…”

“…Fix a minimal idempotent u ∈ E and let C(u) ⊂ X be the dense G δ subset of continuity points of u. Fix some x ∈ X; then, by a theorem of Weiss, P [x], the proximal cell of x, is also a dense G δ subset of X (see [22,Theorem 1.13]). Set A = C(u) ∩ P [x]; then for y ∈ A there is a sequence γ j ∈ Γ such that lim j→∞ γ j x = lim j→∞ γ j y = x.…”

“…Some examples are to be found in Namioka [31] (1984), Milnes [29] (1986) and [30] (1989), Glasner [16] (1976) and [20] (1993), Berg, Gove & Hadad [4] (1998), Budak, Işik, Milnes & Pym [9] (2001), and Glasner & Megrelishvili [23] (2004). Rarely is the enveloping semigroup metrizable (a notable exception is the case of weakly almost periodic metric systems; see Downarowicz [10] (1998) and Glasner [22] (2003), Theorem 1.48).…”