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Register a new userOn $bar{d}$-approachability, entropy density and $mathscr{B}$-free shifts
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We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornsteins $bar{d}$ metric. This leads to a class of shift spaces we call $bar{d}$-approachable. A shift space $bar{d}$-approachable when its canonical sequence of Markov approximations converges to it also in the $bar{d}$ sense. We give a topological characterisation of chain mixing $bar{d}$-approachable shift spaces. As an application we provide a new criterion for entropy density of ergodic measures. Entropy-density of a shift space means that every invariant measure $mu$ of such a shift space is the weak$^*$ limit of a sequence $mu_n$ of ergodic measures with the corresponding sequence of entropies $h(mu_n)$ converging to $h(mu)$. We prove ergodic measures are entropy-dense for every shift space that can be approximated in the $bar{d}$ pseudometric by a sequence of transitive sofic shifts. This criterion can be applied to many examples that were out of the reach of previously known techniques including hereditary $mathscr{B}$-free shifts and some minimal or proximal systems. The class of symbolic dynamical systems covered by our results includes also shift spaces where entropy density was established previously using the (almost) specification property.
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In this work, we preform a systematic investigation about hidden heavy and doubly heavy molecular states from the $D^{(*)}bar{D}^{(*)}/B^{(*)}bar{B}^{(*)}$ and $D^{(*)}D^{(*)}/bar{B}^{(*)}bar{B}^{(*)}$ interactions in the quasipotential Bethe-Salpeter equation (qBSE) approach. With the help of the Lagrangians with heavy quark and chiral symmetries, interaction potentials are constructed within the one-boson-exchange model in which we include the $pi$, $eta$, $rho$, $omega$ and $sigma$ exchanges, as well as $J/psi$ or $Upsilon$ exchange. Possible bound states from the interactions considered are searched for as the pole of scattering amplitude. The results suggest that experimentally observed states, $Z_c(3900)$, $Z_c(4020)$, $Z_b(10610)$, and $Z_b(10650)$, can be related to the $Dbar{D}^{*}$, $D^*bar{D}^{*}$, $Bbar{B}^{*}$, and $B^*bar{B}^{*}$ interactions with quantum numbers $I^G(J^P)=1^+(1^{+})$, respectively. The $Dbar{D}^{*}$ interaction is also attractive enough to produce a pole with $0^+(0^+)$ which is related to the $X(3872)$. Within the same theoretical frame, the existence of $Dbar{D}$ and $Bbar{B}$ molecular states with $0(0^+)$ are predicted. The possible $D^*bar{D}^*$ molecular states with $0(0^+, 1^+, 2^+)$ and $1(0^+)$ and their bottom partners are also suggested by the calculation. In the doubly heavy sector, no bound state is produced from the $DD/bar{B}bar{B}$ interaction while a bound state is found with $0(1^+)$ from $DD^*/bar{B}bar{B}^*$ interaction. The $D^*D^*/bar{B}^*bar{B}^*$ interaction produces three molecular states with $0(1^+)$, $0(2^+)$ and $1(2^+)$.
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Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centraliser and normaliser of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralisers, but large normalisers. In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points and the $k$-free integers in some real quadratic number fields.
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