Abstract
Real numbers can be represented in an arbitrary base β> 1 using the trans- formation Tβ : x → βx −� βxof (0, 1). The closure of the set of the expansions of the real numbers of (0, 1) is called beta-shift. This dynamical system is characterized by the β-expansion of 1. We give a complete classification of the β-expansion of 1 for quartic Pisot units and show that there is no uniform bound on the lengths of the preperiod and the period of the β-expansion of 1.
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