 Skew product

A skew product evolves a base coordinate by a dynamical system $R$ and a fibre coordinate by a map $S_x$ depending on the current base coordinate: $F(x,y)=(R(x),S_x(y))$. The base evolution is independent of the fibre, while the fibre may depend on the base. Continuity or measurability is imposed according to the setting. An irrational skew shift is a useful affine example on a torus.