Rotational beta expansion: ergodicity and soficness. (English) Zbl 1379.37010

Akiyama, Shigeki; Caalim, Jonathan

J. Math. Soc. Japan 69, No. 1, 397-415 (2017).

The authors study a class of piecewise expanding two-dimensional maps \(T.\) These maps generalise the well-known positive and negative \(\beta\)-transformations. It is known that such maps admit an invariant measure \(\mu\) which is absolutely continuous with respect to the two-dimensional Lebesgue measure. The authors give conditions on \(T\) guaranteeing the uniqueness of \(\mu\), and conditions on \(T\) guaranteeing that \(\mu\) is in fact equivalent to the Lebesgue measure.
The authors also study the symbolic dynamical system generated by the map \(T\). In particular, they give conditions ensuring that the associated symbolic dynamical system is sofic.
In the final section of this paper the authors include several useful examples.

Reviewer: Simon Baker (Coventry)

MSC:

37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B10	Symbolic dynamics
37E05	Dynamical systems involving maps of the interval
11K16	Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

Keywords:

beta expansion; invariant measure; sofic system; Pisot number