Alphabets, rewriting trails and periodic representations in algebraic bases

For $beta > 1$ a real algebraic integer ({it the base}), the finite alphabets $mathcal{A} subset mathbb{Z}$ which realize the identity $mathbb{Q}(beta) = {rm Per}_{mathcal{A}}(beta)$, where ${rm Per}_{mathcal{A}}(beta)$ is the set of complex numbers which are $(beta, mathcal{A})$-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base $beta$ and Lehmers problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in $mathbb{Q}(beta)$, generalizing Schmidts theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases $gamma_s := gamma_{n, m_1 , ldots , m_s}$ such that $gamma_{s}^{-1}$ is the unique root in $(0,1)$ of an almost Newman polynomial of the type $-1+x+x^n +x^{m_1}+ldots+ x^{m_s}$, $n geq 3$, $s geq 1$, $m_1 - n geq n-1$, $m_{q+1}-m_q geq n-1$ for all $q geq 1$. For $beta > 1$ a reciprocal algebraic integer close to one, the poles of modulus $< 1$ of the dynamical zeta function of the $beta$-shift $zeta_{beta}(z)$ are shown, under some assumptions, to be zeroes of the minimal polynomial of $beta$.