Mixing actions of the rationals. (English) Zbl 1119.37005

Miles, Richard; Ward, Thomas B.

Ergodic Theory Dyn. Syst. 26, No. 6, 1905-1911 (2006).

Summary: We study mixing properties of algebraic actions of \(\mathbb Q^d\), showing in particular that prime mixing \(\mathbb Q^d\) actions on connected groups are mixing of all orders, as is the case for \(\mathbb Z^d\)-actions. This is shown using a uniform result on the solution of \(S\)-unit equations in characteristic zero fields due to J.-H. Evertse, H.-P. Schlickewei and W. M. Schmidt [Ann. Math. (2) 155, No. 3, 807–836 (2002; Zbl 1026.11038)]. In contrast, algebraic actions of the much larger group \(\mathbb Q^*\) are shown to behave quite differently, with finite order of mixing possible on connected groups.

MSC:

37A25	Ergodicity, mixing, rates of mixing
11D61	Exponential Diophantine equations

Keywords:

mixing properties; algebraic actions

Citations:

Zbl 1026.11038