This paper analyzes price competition in an infinitely repeated duopoly game. In each period, consumers remember the existence and location of their previous supplier. New information is gathered via search or word-of-mouth communication. Market outcomes are history-dependent, and the Markov perfection refinement is used to narrow the set of equilibria. Firms are shown to use mixed pricing strategies in equilibrium. The resulting price dispersion generates non-trivial market share dynamics. The goal of the paper is to characterize these dynamics, and to reveal the driving forces behind them.