An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation (Q2773331)

\textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 8, 477-493 (1957; Zbl 0079.08901)] defined, for every real number \(\beta>1\), the one-sided shift-space \(X_\beta\subset \{0,1,\dots, \lceil\beta\rceil-1\}^{\mathbb{N}}\) corresponding to all \(\beta\)-expansions of real numbers \(x\in [0,1)\), where \(\lceil\beta\rceil\) is the smallest integer greater than or equal to \(\beta\). This space \(X_\beta\) and its properties were investigated by \textit{W. Parry} [Acta Math. Acad. Sci. Hung. 11, 401-416 (1960; Zbl 0099.28103)], where conditions were given for \(X_\beta\) to be sofic or of finite type. \textit{A. Bertrand-Mathis} [Bull. Soc. Math. Fr. 114, 271-323 (1986; Zbl 0628.58024)] and \textit{K. Schmidt} [Bull. Lond. Math. Soc. 12, 269-278 (1980; Zbl 0494.10040)] proved independently that \(X_\beta\) is sofic for Pisot numbers \(\beta>1\) (i.e. for algebraic integers \(\beta>1\) whose conjugates all have moduli strictly less than 1). NEWLINENEWLINENEWLINEThe study of the geometric, arithmetical and symbolic properties of the corresponding two-sided \(\beta\)-shift \(V_\beta\subset \{0,1,\dots, \lceil\beta\rceil\}^{\mathbb{Z}}\) of a Pisot number \(\beta\) was initiated by \textit{A. Vershik} in the special case where \(\beta^2= \beta+1\) [The fibadic expansion of real numbers and adic transformations, Preprint, Mittag-Leffler Institute, 1991/92, Funkt. Anal. Prilozh. 26, 22-27 (1992; Zbl 0810.58031), and St. Petersbg. Math. J. 6, 529-540 (1995); translation from Algebra Anal. 6, No. 3, 94-106 (1994; Zbl 0829.58012)], and extended to all quadratic Pisot units by \textit{N. Sidorov} and \textit{A. Vershik} [J. Dyn. Control Syst. 4, 365-399 (1998; Zbl 0949.37023) and Monatsh. Math. 126, 215-261 (1998; Zbl 0916.28012)]. In these papers the authors exhibit a natural finite-to-one factor map \(\zeta\) from the \(V_\beta\) onto the automorphism \(A\) of the two-torus \(\mathbb{T}^2\) determined by the companion matrix of the minimal polynomial of \(\beta\).NEWLINENEWLINENEWLINEThe connection of this map with the group of homoclinic points of the matrix \(A\), which was implicit in the paper of Vershik and Vershik-Sodorov, was made explicit by \textit{M. Einsiedler} and \textit{K. Schmidt} [Proc. Steklov Inst. Math. 216, 259-279 (1997; Zbl 0954.37008)], and \textit{K. Schmidt} constructed the corresponding map \(\xi:V_\beta\to \mathbb{T}^n\) for Pisot units \(\beta\) of arbitrary degree \(n\) [Monatsh. Math. 129, 37-61 (2000; Zbl 1010.37005); if \(\beta\) is not a unit, the group \(\mathbb{T}^n\) has to be replaced by an appropriate solenoid]. The map \(\xi\) constructed there is, in fact, the restriction of a surjective group homomorphism \(\xi'\) from the group \(\ell^\infty(\mathbb{Z},\mathbb{Z})\) of all bounded two-sided sequences of integers to \(\mathbb{T}^n\) (or to the solenoid). One of the two main conjectures left open in all these papers is that this map is always essentially injective (i.e., injective on a dense \(G_\delta\) set in \(V_\beta\)). NEWLINENEWLINENEWLINEIn the paper by Schmidt it was shown that the conjectured uniqueness of the map \(\zeta\) depended on a certain property of the kernel of the map \(\xi'\) or, equivalently, of the (finite) set of all `tails' of beta-expansions of elements in \(\mathbb{Z}[\beta]\cap [0,1)\). NEWLINENEWLINENEWLINEHere the author investigates this set of `tails' in much greater detail and identifies it with a group naturally associated with the Pisot number \(\beta\) (this is the main contribution of the paper): if NEWLINE\[NEWLINEP_\beta= \{\alpha: \|\alpha\beta^n\|\to 0\text{ as }n\to\infty\} \subset \mathbb{Q}(\beta),NEWLINE\]NEWLINE then \(P_\beta\supset \mathbb{Z}[\beta]\), and the group in question is the quotient \(P_\beta/ \mathbb{Z}[\beta]\). Although this observation, its interpretation and the explicit calculations of these groups in some examples are interesting in their own right, this paper actually sheds no light on the motivating uniqueness problem described above. In a subsequent paper [Bijective and general arithmetic codings for Pisot toral automorphisms, J. Dyn. Control Syst. 7, 447-472 (2001)] \textit{N. Sidorov} shows that the above uniqueness conjecture is equivalent to a further conjecture on Pisot numbers (that every Pisot number is `weakly finitary'), but this conjecture is again left unresolved.