A survey of Max-type recursive distributional equations

In certain problems in a variety of applied probability settings (from probabilistic analysis of algorithms to statistical physics), the central requirement is to solve a recursive distributional equation of the form X-d = g((xi(i), X-i), i >= 1). Here (xi(i)) and g((.)) are given and the X-i are independent copies of the unknown distribution X. We survey this area, emphasizing examples where the function g((.)) is essentially a "maximum" or "minimum" function. We draw attention to the theoretical question of endogeny: in the associated recursive tree process X-i, are the X-i measurable functions of the innovations process (xi(i))?