By trace map we mean the following polynomial map of R^3:
T(x,y,z)= (2xy-z, x, y).

Despite of its simple form, it is related to complicated mathematical objects such as character varieties of some surfaces, Painlev\'e sixth equation, and discrete Schr\"odinger operator with Fibonacci potential. We will present some very recent results on dynamics of the trace map and discuss their applications. These is a joint project with D.Damanik.