β-Expansions and symbolic dynamics

Dynamics forβ-shifts and Diophantine approximation

“…In this section, we give a continuum of explicit transcendental numbers lying in Class C 4 of the classification of β-shifts introduced by Blanchard in [12]. We first briefly recall this classification and some related results.…”

“…At the end of [12], Blanchard asked whether there are transcendental numbers in every Class C 3 , C 4 and C 5 . A partial answer is given by Allouche, Bertrand-Mathis and Mauduit in the Appendix of the same paper (related results are recalled below).…”

“…The special case where u n = 2 n−1 is suggested by Mauduit in [12] as an example providing a non-specified subshift associated to a transcendental number. The transcendence of the associated β follows from a classical result by Mahler [27].…”

The spectrum of Poincaré recurrence

“…Clearly each map is positively expansive. The following results are known for T β [4,28]: (i) the set of β such that T β admits a finite Markov partition is at most countable; and (ii) the set of β such that T β satisfies the SPEC has Hausdorff dimension 1 and Lebesgue measure 0.…”

“…For fixed n ∈ N, let {n k }, {N k } be the two sequences in the Moran set F as given in Proposition 4.6; for example, we can take n k = n(k + k 0 ) 4 and N k = [e (k+k 0 ) 4 ],…”