Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval

Abstract

The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces (Xi)∞i=1 and a sequence of continuous maps (fi)∞i=1, fi : Xi → Xi+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 ○ . . . ○ f2 ○ f1. As an application we construct a large class of smooth triangular maps of the square of type 2∞ and positive topological entropy.