The local linearization of fixed points is one of the central problems in dynamical systems. In the case of complex quadratic polynomials, Yoccoz proved that small cycles are the only obstruction to the linearization of irrationally indifferent fixed points. In positive characteristic, there are no known examples of univariate polynomial maps with small cycles. After an overview of these considerations, the talk will focus on the connection between small cycles in positive characteristic and the lower ramification numbers of wildly ramified power series.