Contribution on The Intrinsic Ergodicity of the Negative Beta-shift

Let $\beta$ be a real number less than -1. In this paper, we prove the uniqueness of the measure with maximal entropy of the negative $\beta$-shift. Endowed with the shift, this symbolic dynamical system is coded under certain conditions, but in all cases, it is shown that the measure with maximal entropy is carried by a support coded by a recurrent positive code. One of the difference between the positive and the negative $\beta$-shift is the existence of gaps in the system for certain negative values of $\beta$ . These are intervals of negative $\beta$-representations (cylinders) negligible with respect to the measure with maximal entropy, which is a measure of Champernown.