Amoebas are geometric objects attached to polynomials or polynomial ideals in several variables. Although the term was coined only in 1994, the concept has much earlier roots. I'll descibe how amoebas arise in studying the expansive behavior of the joint action of several commuting automorphisms of a compact abelian group. I'll also show how the dynamical idea of homoclinic point gives a simple explanation for a property of the union of the p-adic amoebas of a fixed polynomial with integer coefficients, leading to one aspect of a local-global principle. There is much overlap with the developing area of "tropical algebraic geometry".