Critical exponent of infinite words coding beta-integers associated with non-simple Parry numbers

In this paper, we study the critical exponent of infinite words $\text{ubeta}$ coding $\beta$-integers for $\beta$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the infinite word in question. We calculate also the ultimate critical exponent that expresses how long repetitions occur in the infinite word $\text{ubeta}$ when the factors of length growing ad infinitum are considered. The basic ingredients of our method are the description of all bispecial factors of $\text{ubeta}$ and the notion of return words. This method can be applied to any fixed point of any primitive substitution.