(In collaboration with K. Frączek, J. Kułaga-Przymus and M. Lemańczyk)
An automorphism $ T $ of a probability space hasasymptotic orthogonal
powers if all
ergodic joinings of the the product $ T ^ r \otimes T ^ s $ converge to
the product joining, when the relatively
prime integers $ r $ and $ s $ go to infinity.
We show that the affine unipotent and ergodic diffeomorphisms of
nilmanifolds
have asymptotic orthogonal powers. Two consequences follow:
- The conjecture of Sarnak on the orthogonality of the Möbius function
to deterministic systems
is true for any automorphism measurably isomorphic to an affine unipotent
and ergodic diffeomorphism of a nilmanifold.
- In addition, for these automorphims,
the above mentioned Sarnak conjecture "holds on small intervals".