Criterium involving ergodicity and weak mixing

A transformation TT on XX is ergodic iff for any two measurable sets
UU and VV holds: limn→∞1n∑n−1j=0m(T−jU∩V)=m(U)m(V)\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} m(T^{-j}U\cap V)=m(U)m(V),

(or equivalently iff every invariant measurable
function is constant almost everywhere or iff every T-invariant set has full measure or measure 0).

A transformation TT is called weakly mixing if for any two measurable sets
UU and VV, \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} |m(T^{-j}U\cap V)-m(U)m(V)|=0\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1} |m(T^{-j}U\cap V)-m(U)m(V)|=0.

How to prove that the following properties are equivalent:

1) TT is weakly mixing,

2) T\times TT\times T is ergodic with respect to m\times mm\times m,

3) T\times TT\times T is weakly mixing with respect to m\times mm\times m.

3) implies 2), but how to prove the rest?

Any help is welcome. Thanks in advance.



1 Answer


I think this proposition is very standard in Ergodic Theory textbooks, like Walters’ Introduction to Ergodic Theory, but I assume that is not what you want.

Some directions: to prove that weak mixing of the system implies weak mixing for the product, try to prove it for the algebra generated by products of measurable sets using certain inequality which is recurring in analysis.

To prove that ergodicity of the product implies weak mixing of the original system, you may use the following proposition:

For a bounded sequence (a_n)(a_n) of real numbers, then \lim_n\dfrac{1}{n}\sum_{j=1}^n|a_n|=0 \text{ if and only if } \lim_n\dfrac{1}{n}\sum_{j=1}^n(a_n)^2=0. \lim_n\dfrac{1}{n}\sum_{j=1}^n|a_n|=0 \text{ if and only if } \lim_n\dfrac{1}{n}\sum_{j=1}^n(a_n)^2=0.