Deterministic functions on amenable semigroups and a generalization of the Kamae-Weiss theorem on normality preservation

Hereditary subshifts whose measure of maximal entropy does not have the Gibbs property

“…To prove Proposition 3.8, we will need a certain "lifting lemma" from [3] for quasi-generic points to joinings. We formulate it here for Z-actions, while the original version is more general (the result is true for actions of countable cancellative semigroups and arbitrary Følner sequences, see [3]).…”

On arithmetic functions orthogonal to deterministic sequences

“…Using Lemma 6.4, find a joining of κ and ν for which π 0 (in L 2 pX u , κq) is not orthogonal to π 0 in L 2 pνq: ş π 0 b π 0 dρ ‰ 0. Now, use a subsequence version of the lifting lemma (Theorem 5.16 in [6]) to find y in the subshift defining the Bernoulli automorphism such that pu, yq is generic, along a subsequence pN k q " pM k q, for ρ. Then…”

“…Definition 1.2. Given a characteristic class F we say that a (bounded) arithmetic function u : N Ñ C satisfies the Veech condition with respect to F if (6) π 0 K L 2 ppBpX u q, κq F q for each Furstenberg system κ P V S puq.…”

“…Our first proof of Theorem A was based on a different lifting lemma by Bergelson, Downarowicz and Vandehey (Theorem 5.16 in [6]) and a recent result by Downarowicz and Weiss showing the existence, for each zero-entropy measure-theoretic system, of a special Hansel model which is symbolic [13]. The proof we finally chose to present here does not require the use of this symbolic model, as our lifting lemma works for all topological systems.…”

“…DDKBSZ stands for Daboussi-Delange-Kátai-Bourgain-Sarnak-Ziegler[7],[35] 6. In fact, it is open whether a non-periodic, zero entropy, continuous, algebraic automorphism of T 2 satisfies the strong µ-MOMO property.…”