Abstract. Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences.

Given B ⊂ N \mathscr {B}\subset \mathbb {N} , let η = η B ∈ { 0 , 1 } Z \eta =\eta _{\mathscr {B}}\in \{0,1\}^{\mathbb {Z}} be the characteristic function of the set F B := Z ∖ ⋃ b ∈ B b Z \mathcal {F}_\mathscr {B}:=\mathbb {Z}\setminus \bigcup _{b\in \mathscr {B}}b\mathbb {Z} of B \mathscr {B} -free numbers. The B \mathscr {B} -free shift ( X η , S ) (X_\eta ,S) , its hereditary closure ( X ~ η , S ) (\widetilde {X}_\eta ,S) , and (still larger) the B \mathscr {B} -admissible shift ( X B , S ) (X_{\mathscr {B}},S) are examined. Originated by Sarnak in 2010 for B \mathscr {B} being the set of square-free numbers, the dynamics of B \mathscr {B} -free shifts was discussed by several authors for B \mathscr {B} being Erdös; i.e., when B \mathscr {B} is infinite, its elements are pairwise coprime, and ∑ b ∈ B 1 / b > ∞ \sum _{b\in \mathscr {B}}1/b>\infty : in the Erdös case, we have X η = X ~ η = X B X_\eta =\widetilde {X}_\eta =X_{\mathscr {B}} . It is proved that X η X_\eta has a unique minimal subset, which turns out to be a Toeplitz dynamical system. Furthermore, a B \mathscr {B} -free shift is proximal if and only if B \mathscr {B} contains an infinite coprime subset. It is also shown that for B \mathscr {B} with light tails, i.e., d ¯ ( ∑ b > K b Z ) → 0 \overline {d}(\sum _{b>K}b\mathbb {Z})\to 0 as K → ∞ K\to \infty , proximality is the same as heredity. For each B \mathscr {B} , it is shown that η \eta is a quasi-generic point for some natural S S -invariant measure ν η \nu _\eta on X η X_\eta . A special role is played by subshifts given by B \mathscr {B} which are taut, i.e., when δ ( F B ) > δ ( F B ∖ { b } ) \boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}})>\boldsymbol {\delta }(\mathcal {F}_{\mathscr {B}\setminus \{b\}}) for each b ∈ B b\in \mathscr {B} ( δ \boldsymbol {\delta } stands for the logarithmic density). The taut class contains the light tail case; hence all Erdös sets and a characterization of taut sets B \mathscr {B} in terms of the support of ν η \nu _\eta are given. Moreover, for any B \mathscr {B} there exists a taut B ′ \mathscr {B}’ with ν η B = ν η B ′ \nu _{\eta _{\mathscr {B}}}=\nu _{\eta _{\mathscr {B}’}} . For taut sets B , B ′ \mathscr {B},\mathscr {B}’ , it holds that X B = X B ′ X_\mathscr {B}=X_{\mathscr {B}’} if and only if B = B ′ \mathscr {B}=\mathscr {B}’ . For each B \mathscr {B} , it is proved that there exists a taut B ′ \mathscr {B}’ such that ( X ~ η B ′ , S ) (\widetilde {X}_{\eta _{\mathscr {B}’}},S) is a subsystem of ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) and X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} is a quasi-attractor. In particular, all invariant measures for ( X ~ η B , S ) (\widetilde {X}_{\eta _{\mathscr {B}}},S) are supported by X ~ η B ′ \widetilde {X}_{\eta _{\mathscr {B}’}} . Moreover, the system ( X ~ η , S ) (\widetilde {X}_\eta ,S) is shown to be intrinsically ergodic for an arbitrary B \mathscr {B} . A description of all probability invariant measures for ( X ~ η , S ) (\widetild

Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form [N, N + √ N) also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.

We show that Sarnak's conjecture on Möbius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial P ∈ R[x] with irrational leading coefficient and for each multiplicative function ν : N → C, |ν| ≤ 1, we have 1 M M ≤m<2M 1 H m≤n

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in {−1, 0, 1} N * , and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.