Abstract:

The widely studied class of β-transformations (T_β (x) = βx mod 1, x \in [0,1), β \in (1,\infty)) is known to generate a transitive coded shift by the coding of itineraries of each point in the interval. We call it β-shift. It was proved in [Par] and [BM] that the orbit of 1 under T_β determines the class of the generated β-shift, that is, whether it is of finite type, sofic, specified, synchronized or none of these. Recently, Díaz, Gelfert and Rams proposed a model of a partially hyperbolic skew-product with a pair of concave interval maps on the
fiber. They proved in [DGR] that the language collecting all possible concatenations of these concave maps generates an essentially coded shift. Inspired by the known results on β-shifts, we propose, in the
[DGR] setting, to establish a relation between the orbits of the interval’s extremal points under the concave pair and the class of the generated shift.

[DGR] Díaz, Gelfert, Rams. Mingled Hyperbolicities: Ergodic Properties and Bifurcation Phenomena (An Approach Using Concavity). Discrete and Continuous Dynamical Systems. 2022

[BM] Bertrand-Mathis. Specification, Synchronisation, Average Length. Bull. Soc. Math. Fr. 1986

[Par] Parry. On The β-Expansions of Real Numbers. Acta Math. Acad. Sci. Hung. 1960