arXiv subjects
Publications and source records attributed to Hiromi Ei.
We give a concrete construction of a natural extension of $(-β)$-transformation when $β$ is greater than the golden mean. Our construction relies on its Markov diagram and the eigenvectors of the associated countable Markov shifts. Its positive recurrence can be shown by path counting using a special property of the diagram. Our down-to-earth construction elucidates the result of Bruin-Kalle \cite{BruinKalle} by examples.
For substitution satisfying Pisot, irreducible, unimodular condition, a tiling substitution plays a key role in construction of a stepped surface and Rauzy fractal. In this paper we will extend the method to hyperbolic automorphisms on the free group of rank 2 in some class, and obtain set equations of Rauzy fractals by virtue of a tiling substitution. We will also see that the domain exchange transformation on Rauzy fractal is just a two interval exchange transformation.