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

For each B-free subshift given by B = {2 i bi} i∈N , where {bi} i∈N is a set of pairwise coprime odd numbers greater than one, it is shown that the automorphism group of the subshift consists solely of powers of the shift.

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We characterize proximality of multidimensional B-free systems in the case of number fields and lattices in Z m , m ě 2.1. Introduction. In this paper, we study proximality of generalizations of B-free systems [1,10]. Let B Ď N. Integers with no factors in B are called B-free numbers and are denoted by F B . Such sets were studied already in the 30's by Behrend, Chowla, Davenport, Erdős, Schur and others. Note that, if S " tp 2 : p is primeu then 1 F S " µ 2 , where µ : Z Ñ C is the Möbius function given by the following formula:The dynamical approach to study B-free systems is rather new. In 2010, Sarnak in his seminal paper [17] proposed to study the dynamical systems determined by µ and µ 2 . In either case, we consider the closure X η of the orbit of η " µ P t´1, 0, 1u Z or η " µ 2 P t0, 1u Z under the left shift S. The dynamics of pX µ , Sq is complicated and there are many open questions related to it. The system pX µ 2 , Sq (called squarefree system) which is a topological factor of pX µ , Sq via the map px n q nPZ Þ Ñ px 2 n q nPZ is simpler to study. Similarly, given B Ď N, taking the closure of the orbit of η " 1 F B P t0, 1u Z under the left shift, yields a B-free system. At first, B-free systems were studied in the Erdős case, i.e., for B infinite, pairwise coprime, withLet G be a countable abelian group. Recall that proper subgroups H 1 , H 2 Ď G are said to be coprime whenever H 1`H2 " G. Let pT g q gPG be an action of G by homeomorphisms on a compact metric space pX, Dq. The pair pX, pT g q gPG q is called a topological dynamical system. Two complementary, basic concepts of topological dynamics are distality and proximality. Recall that a pair px, yq of points from X is called distal if lim inf gÑ8 DpT g x, T g yq ą 0, otherwise px, yq is called proximal. If any pair of distinct points in X is distal (respectively, proximal) then the