The natural extension of the $(-\beta )$-transformation

We give a concrete construction of a natural extension of $(-\beta )$-transformation when $\beta$ 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.