Intrinsic Ergodicity of the negative beta-shift

Florent Nguema-Ndong

Published 2020-05-05Version 1

Let $ \beta $ be a real less than -1. In this paper, we prove the unicity 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 case, 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 $ . It is about of intervals of negative $\beta$-representations (cylinders) negligible with respect to the measure with maximal entropy. Nervertheless, this measure is a measure of Champernown.