Dynamical properties of minimal Ferenczi subshifts

Abstract

We provide an explicit $\mathcal {S}$ -adic representation of rank-one subshifts with bounded spacers and call the subshifts obtained in this way ‘minimal Ferenczi subshifts’. We aim to show that this approach is very convenient to study the dynamical behavior of rank-one systems. For instance, we compute their topological rank, the strong and the weak orbit equivalence class. We observe that they have an induced system that is a Toeplitz subshift having discrete spectrum. We also characterize continuous and non-continuous eigenvalues of minimal Ferenczi subshifts.

The Jacobs–Keane theorem from the $ \mathcal{S} $-adic viewpoint

“…, n ≥ 0 and hence from Item 6 in Theorem 1.2 we obtain that this induced system is a regular extension of its maximal equicontinuous topological factor. This improves the remark made in [2] saying that these systems are mean equicontinuous (we refer to [34] for the definition and details about this notion).…”

“…We refer to [28,29] for a more extensive discussion on these subshifts. Minimal rank-one subshifts are studied from the Sadic perspective in [2], where the authors prefer to call them Ferenczi subshifts. In [2, Section 4.3] it is shown that a minimal Ferenczi subshift possesses an induced system that is (conjugate to) a Toeplitz subshift generated by the constant length directive sequence τ = (τ n : A * n+1 → A * n ) n≥0 which have the form τ n (a) = L n aR n , a ∈ A n+1 (34) for some nonempty words L n , R n in A * n .…”

“…Introduction. A series of recent works shows that the underlying S-adic structures of zero-entropy subshifts shed new light on the study of their properties as shown by [11,9,17,27,25,35,2,26]. This approach, which goes back to [47,28], consists in finding a representation of the subshift in terms of an infinite sequence of morphisms defined on finitely generated monoids.…”