Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces (X i) ∞ i=1 and a sequence of continuous maps (f i) ∞ i=1 , f i : X i → X i+1 , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of fn •. .. • f 2 • f 1. As an application we construct a large class of smooth triangular maps of the square of type 2 ∞ and positive topological entropy.