LIFTS OF ONE-DIMENSIONAL SYSTEMS .1. HYPERBOLIC BEHAVIOR

We define the n-lift of a one-dimensional system x(i+1) = f(x(i)). The n-lift can be thought of as a perturbation of the one-dimensional sys tem depending on the state of the system n - 1 time-steps back. We pro ve that certain f-invariant Canter sets give invariant Canter sets in the lifted system. We prove that if f has an invariant hyperbolic Cant er set then the lifted system has an invariant hyperbolic Canter set p rovided the derivatives of f obey a simple condition. We also prove th at hyperbolicity is preserved if the same conditions on the derivative s of f hold.