Dynamics of quadratic polynomials over local fields (Q933194)

The authors determine the structure of the Julia set for quadratic polynomials over a local field. Let \(K\) be a field complete under a rank one valuation \(v\), let \(P(X)=X^2+bX+c\in K[X]\) and put \(\Delta=(b-1)^2-4c\). Moreover denote by \(C_v\) the completion of the algebraic closure of \(K\). It is shown that if for all \(\xi\in K\) one has \(v(P(\xi)-\xi)>1\), then the set \(J_P(K)=J_P(C_v)\cap P^1(K)\) of \(K\)-points of the Julia set and the filled Julia set \(JR_P(K)\) of \(P\) are both empty. If for some \(\xi\in K\) one has \(v(P(\xi)-\xi)\leq 1\), then there are two cases: if \(v(\Delta)\leq1\), then \(J_P(K)\) is empty and \(JR_P(K)\) is a closed ball of radius one and center \(\xi\), and otherwise \(J_P(K)=JR_P(K)\) is a non-empty compact set on which \(P\) is topologically conjugated to the one-sided shift on two symbols.