For an arbitrary set containing 1, an ergodic automorphism T whose set of essential values of the multiplicity function is equal to A is constructed. If A is additionally finite, T can be chosen to be an analytic diffeomorphism on a finite-dimensional torus.
For an arbitrary set A ⊆ ℕ satisfying 1 ∈ A and lcm(m₁,m₂) ∈ A whenever m₁,m₂ ∈ A, an ergodic abelian group extension of a rotation for which the range of the multiplicity function equals A is constructed.
If is a sequence of real numbers which is good for the ergodic theorem, is the sequence of the integer parts good for the ergodic theorem ? The answer is negative for the mean ergodic theorem and affirmative for the pointwise ergodic theorem.
The uniqueness theorem for the ergodic maximal operator is proved in the continous case.
It is proved that the ergodic maximal operator is one-to-one.
It is shown that if two functions share the same uncentered (two-sided) ergodic maximal function, then they are equal almost everywhere.